UNIT 4: ADVANCED LINEAR ALGEBRA - EXAM-FOCUSED SHORT NOTES
I. FOUNDATIONS: SUBSPACES, SUMS, AND QUOTIENTS
Direct Sum of Subspaces
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Definition: For subspaces \( U_1, \dots, U_k \) of \( V \), the sum \( U_1 + \cdots + U_k \) is a direct sum, denoted \( V = U_1 \oplus \cdots \oplus U_k \), if every \( v \in V \) can be written uniquely as \( v = u_1 + \cdots + u_k \) with \( u_i \in U_i \).
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Equivalent Condition: \( U_1 + \cdots + U_k \) is direct iff \( U_i \cap (U_1 + \cdots + U_{i-1} + U_{i+1} + \cdots + U_k) = \{0\} \) for all \( i \). For two subspaces: \( U_1 \oplus U_2 \) iff \( U_1 \cap U_2 = \{0\} \).
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Dimension Formula (Two Subspaces):
$$ \dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2). $$
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Example (Non-Direct Sum):
Let \( U_1 = \{(x,y,0)\} \), \( U_2 = \{(0,0,z)\} \), \( U_3 = \{(0,y,y)\} \) in \( F^3 \). Then \( U_1 + U_2 + U_3 = F^3 \), but not direct because \( (0,1,0) \in U_1 \cap (U_2 + U_3) \) nontrivially.
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[!TIP]
To prove a sum is not direct, find a nonzero vector with two distinct representations or show \( U_i \cap (\sum_{j\neq i} U_j) \neq \{0\} \).
Quotient Spaces
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Definition: For subspace \( W \subseteq V \), the quotient space \( V/W = \{ v + W \mid v \in V \} \), where \( v+W \) is the coset (equivalence class modulo \( W \)).
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Operations:
\( (v+W) + (u+W) = (v+u)+W \),
\( \alpha(v+W) = (\alpha v)+W \).
These are well-defined because \( W \) is a subspace.
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Key Theorem: If \( V \) is finite-dimensional, then
$$ \dim(V/W) = \dim V - \dim W. $$
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Proof Sketch: Extend a basis \( \{w_1,\dots,w_m\} \) of \( W \) to a basis \( \{w_1,\dots,w_m, v_1,\dots,v_n\} \) of \( V \). Then \( \{v_1+W, \dots, v_n+W\} \) is a basis of \( V/W \).
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[!TIP]
Quotient spaces are always vector spaces. To show \( V/W \) is finitely generated, take a finite generating set for \( V \) and map to \( V/W \).
Annihilator of a Subspace
- Definition: For subspace \( W \subseteq V \), the annihilator is
$$ W^0 = \{ f \in V^* \mid f(w)=0 \; \forall w \in W \}, $$
where \( V^* \) is the dual space.
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Properties:
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\( W^0 \) is a subspace of \( V^* \).
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Dimension Formula: \( \dim W + \dim W^0 = \dim V \) (for finite-dimensional \( V \)).
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Double Annihilator: \( (W^0)^0 = W \) (for finite-dimensional \( V \)).
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Proof of Dimension Formula: Let \( \dim V = n \), \( \dim W = m \). Extend basis \( \{w_1,\dots,w_m\} \) of \( W \) to basis \( \{w_1,\dots,w_m, v_1,\dots,v_{n-m}\} \) of \( V \). Let \( \{f_1,\dots,f_n\} \) be dual basis. Then \( W^0 = \operatorname{span}\{f_{m+1},\dots,f_n\} \), so \( \dim W^0 = n-m \).
II. DUAL SPACES & LINEAR FUNCTIONALS
Dual Space \( V^* \)
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Definition: \( V^* = \mathcal{L}(V, F) \), the space of all linear functionals \( f: V \to F \). It is a vector space over \( F \) with usual operations.
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Dual Basis: If \( \mathcal{B} = \{v_1,\dots,v_n\} \) is a basis of \( V \), the dual basis \( \mathcal{B}^* = \{f_1,\dots,f_n\} \subseteq V^* \) satisfies \( f_i(v_j) = \delta_{ij} \). Every \( f \in V^* \) writes uniquely as \( f = \sum_{i=1}^n f(v_i) f_i \).
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Double Dual: \( V^{**} = (V^*)^* \). The map \( \eta: V \to V^{**} \) defined by \( \eta(v)(f) = f(v) \) is a linear isomorphism for finite-dimensional \( V \). Thus \( V \cong V^{**} \) naturally.
Representation of Linear Functionals
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Riesz Representation Theorem for \( \mathbb{R}^n \):
For any linear functional \( f \) on \( \mathbb{R}^n \), there exists a unique vector \( \bar{a} \in \mathbb{R}^n \) such that
$$ f(\bar{r}) = \bar{r} \cdot \bar{a} \quad \forall \bar{r} \in \mathbb{R}^n. $$
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Proof: Let \( \mathcal{E} \) be standard basis. Define \( a_i = f(e_i) \). Then for \( \bar{r} = \sum r_i e_i \),
\( f(\bar{r}) = \sum r_i f(e_i) = \sum r_i a_i = \bar{r} \cdot \bar{a} \). Uniqueness: if \( \bar{r} \cdot \bar{a} = 0 \) for all \( \bar{r} \), then \( \bar{a}=0 \).
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Generalization: In an inner product space \( V \), for every \( f \in V^* \), there exists a unique \( y \in V \) such that \( f(x) = \langle x, y \rangle \) for all \( x \in V \).
III. LINEAR TRANSFORMATIONS: INVARIANCE & DECOMPOSITION
Invariant Subspaces
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Definition: A subspace \( W \subseteq V \) is \( T \)-invariant if \( T(W) \subseteq W \).
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Examples:
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Eigenspaces: \( E_\lambda = \ker(T - \lambda I) \) are \( T \)-invariant.
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\( \{0\} \) and \( V \) are always invariant.
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For \( T \) diagonalizable, any sum of eigenspaces is invariant.
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[!TIP]
To find invariant subspaces, look for kernels or images of polynomials in \( T \), or subspaces spanned by eigenvectors.
Primary Decomposition Theorem
- Statement: Let \( T: V \to V \) be linear, \( \dim V < \infty \). Let the minimal polynomial be
$$ p_T(t) = p_1(t)^{e_1} p_2(t)^{e_2} \cdots p_k(t)^{e_k}, $$
where \( p_i(t) \) are distinct monic irreducible polynomials over \( F \). Then
$$ V = \ker p_1(T)^{e_1} \oplus \ker p_2(T)^{e_2} \oplus \cdots \oplus \ker p_k(T)^{e_k}. $$
Each \( W_i = \ker p_i(T)^{e_i} \) is \( T \)-invariant, and the minimal polynomial of \( T|_{W_i} \) is \( p_i(t)^{e_i} \).
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Significance: Decomposes \( V \) into \( T \)-invariant subspaces where \( T \) acts "independently" with minimal polynomial a power of an irreducible.
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Application: Given \( p_T \), find primary components. For example, if \( p_T(t) = (t-1)^2 (t-2) \), then \( V = \ker(T-I)^2 \oplus \ker(T-2I) \).
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Proof Sketch: Since \( p_i \) are pairwise coprime, there exist polynomials \( q_i \) such that \( \sum q_i p_i^{e_i} = 1 \). Define projection \( P_i = q_i(T) p_i(T)^{e_i} \). Then \( P_i \) projects onto \( W_i \), \( P_i P_j = 0 \) for \( i \neq j \), \( \sum P_i = I \), and \( P_i(V) = W_i \). Hence direct sum.
IV. EIGENVALUES, EIGENVECTORS & DIAGONALIZATION
Characteristic Polynomial & Eigenvalues
- Definition: For \( A \in M_n(F) \), the characteristic polynomial is
$$ p_A(t) = \det(tI - A). $$
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Eigenvalues: Roots of \( p_A(t) = 0 \). For \( \lambda \) eigenvalue, \( \ker(A - \lambda I) \neq \{0\} \).
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Theorem: Eigenvalues of a triangular matrix are its diagonal entries.
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[!TIP]
Always compute \( p_A(t) \) correctly: \( \det(tI - A) \), not \( \det(A - tI) \) (sign difference for odd \( n \)).
Algebraic vs. Geometric Multiplicity
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Algebraic Multiplicity (AM): Multiplicity of \( \lambda \) as root of \( p_A(t) \).
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Geometric Multiplicity (GM): \( \dim \ker(A - \lambda I) \), i.e., number of linearly independent eigenvectors for \( \lambda \).
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Inequality:
$$ 1 \leq \text{GM} \leq \text{AM}. $$
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[!TIP]
GM is the number of Jordan blocks for \( \lambda \). If GM < AM, \( A \) is not diagonalizable.
Diagonalizability
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Criterion: \( A \) is diagonalizable iff:
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Sum of geometric multiplicities equals \( n \), or
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Minimal polynomial splits into distinct linear factors (i.e., no repeated roots).
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Theorem: If \( T \) has \( n \) distinct eigenvalues, then \( T \) is diagonalizable.
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Process:
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Find eigenvalues \( \lambda_i \).
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For each \( \lambda_i \), find basis of eigenspace \( \ker(A - \lambda_i I) \).
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If total vectors = \( n \), then \( A \) is diagonalizable. Form \( P \) with eigenvectors as columns, then \( P^{-1}AP = D \) (diagonal with eigenvalues).
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Example:
\( A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \).
\( p_A(t) = (3-t)^2 - 1 = t^2 - 6t + 8 = (t-2)(t-4) \).
Eigenvalues \( 2,4 \) (distinct) ⇒ diagonalizable.
For \( \lambda=2 \): \( (A-2I) = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \), eigenvector \( (1,-1) \).
For \( \lambda=4 \): \( (A-4I) = \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix} \), eigenvector \( (1,1) \).
\( P = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \), \( D = \begin{bmatrix} 2 & 0 \\ 0 & 4 \end{bmatrix} \).
Cayley-Hamilton Theorem
- Statement: Every square matrix satisfies its own characteristic equation:
$$ p_A(A) = 0. $$
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Proof Sketch (via Adjugate):
Let \( B(t) = \operatorname{adj}(tI - A) \). Then \( (tI - A) B(t) = p_A(t) I \). Each entry of \( B(t) \) is a polynomial in \( t \) of degree ≤ \( n-1 \). Write \( B(t) = B_{n-1} t^{n-1} + \cdots + B_0 \). Substitute \( t=A \):
\( (A I - A) B(A) = p_A(A) I \) ⇒ \( 0 = p_A(A) I \), so \( p_A(A)=0 \).
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Applications:
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Compute \( A^k \) for \( k \geq n \) by reducing powers using \( p_A(A)=0 \).
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Find \( A^{-1} \): If \( p_A(t) = t^n + c_{n-1}t^{n-1} + \cdots + c_0 \), then
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$$ A^{-1} = -\frac{1}{c_0} (A^{n-1} + c_{n-1}A^{n-2} + \cdots + c_1 I) \quad \text{if } c_0 \neq 0. $$
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Example (Dec 2024):
\( B = \begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix} \), \( p_B(t) = t^2 - 9t + 14 \).
Then \( B^2 - 9B + 14I = 0 \) by Cayley-Hamilton.
V. CANONICAL FORMS: JORDAN & RATIONAL
Jordan Canonical Form (JCF)
- Jordan Block: \( J(\lambda, k) \) is a \( k \times k \) matrix with \( \lambda \) on diagonal, 1's on superdiagonal, 0 elsewhere:
$$ J(\lambda,k) = \begin{bmatrix} \lambda & 1 & & \\ & \lambda & \ddots & \\ & & \ddots & 1 \\ & & & \lambda \end{bmatrix}. $$
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Definition: A matrix is in Jordan canonical form if it is block diagonal with Jordan blocks \( J(\lambda_i, k_i) \). The eigenvalues \( \lambda_i \) may repeat.
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Existence & Uniqueness: Over an algebraically closed field (e.g., \( \mathbb{C} \)), every matrix is similar to a unique JCF (up to block ordering).
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Computing JCF:
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Find eigenvalues \( \lambda \) from \( p_A(t) \). For each \( \lambda \), compute AM and GM.
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For each \( \lambda \), compute sequence \( d_k = \dim \ker(A - \lambda I)^k \) for \( k=1,2,\dots \) until \( d_k = \text{AM} \).
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Number of Jordan blocks of size \( \geq k \) is \( d_k - d_{k-1} \) (with \( d_0=0 \)).
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Determine block sizes from these numbers.
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Construct Jordan basis using generalized eigenvectors: for each block of size \( k \), find chain \( v, (A-\lambda I)v, \dots, (A-\lambda I)^{k-1}v \).
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Example (Jun 2025):
\( A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix} \).
\( p_A(t) = (t-1)^2(t-3) \).
For \( \lambda=1 \):
\( A-I = \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{bmatrix} \), \( \ker(A-I) = \operatorname{span}\{(0,1,0)\} \), so \( d_1=1 \).
\( (A-I)^2 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 4 \end{bmatrix} \), \( \ker(A-I)^2 = \operatorname{span}\{(1,0,0),(0,1,0)\} \), so \( d_2=2=\text{AM} \).
Then \( d_1-d_0=1 \) block of size ≥1, \( d_2-d_1=1 \) block of size ≥2 ⇒ one block of size 2.
For \( \lambda=3 \): \( A-3I \) has kernel dimension 1, AM=1 ⇒ one block of size 1.
JCF: \( \operatorname{diag}(J_2(1), J_1(3)) = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix} \).
Rational Canonical Form (Brief)
- Companion Matrix: For monic polynomial \( p(t) = t^k + a_{k-1}t^{k-1} + \cdots + a_0 \), the companion matrix is
$$ C(p) = \begin{bmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{k-1} \end{bmatrix}. $$
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Rational Canonical Form: Every matrix is similar to a block diagonal matrix of companion matrices of invariant factors (divisibility chain). Unique over any field.
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[!TIP]
JCF requires algebraically closed field; rational canonical form exists over any field.
VI. INNER PRODUCT SPACES & ADJOINTS
Inner Product Spaces
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Definition (Complex): An inner product on vector space \( V \) over \( F \) (\( \mathbb{R} \) or \( \mathbb{C} \)) is a function \( \langle \cdot, \cdot \rangle: V \times V \to F \) satisfying:
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Sesquilinearity: \( \langle \alpha x + \beta y, z \rangle = \overline{\alpha}\langle x,z \rangle + \overline{\beta}\langle y,z \rangle \), and linear in second argument (or conjugate linear in first—convention varies; we use linear in second).
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Conjugate Symmetry: \( \langle x, y \rangle = \overline{\langle y, x \rangle} \).
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Positive Definiteness: \( \langle x, x \rangle > 0 \) for \( x \neq 0 \).
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Real Case: All conjugates drop, so symmetric and bilinear.
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Norm: \( \|x\| = \sqrt{\langle x, x \rangle} \).
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Examples:
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\( \mathbb{R}^n \) with dot product \( \bar{x} \cdot \bar{y} = \sum x_i y_i \).
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\( \mathbb{C}^n \) with \( \langle \bar{x}, \bar{y} \rangle = \sum x_i \overline{y_i} \).
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\( C[a,b] \) with \( \langle f,g \rangle = \int_a^b f(x)\overline{g(x)} dx \).
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[!TIP]
To show a given form is an inner product, verify all three properties. For positive definiteness, check \( \langle x,x \rangle \geq 0 \) and \( =0 \iff x=0 \).
Orthogonality & Gram-Schmidt
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Orthogonal Set: \( \{v_1,\dots,v_k\} \) with \( \langle v_i, v_j \rangle = 0 \) for \( i \neq j \).
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Orthonormal Set: Orthogonal and \( \|v_i\|=1 \).
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Gram-Schmidt Orthogonalization:
Given linearly independent \( \{u_1,\dots,u_n\} \), define:
\( v_1 = u_1 \),
\( v_2 = u_2 - \frac{\langle u_2, v_1 \rangle}{\langle v_1, v_1 \rangle} v_1 \),
\( v_3 = u_3 - \frac{\langle u_3, v_1 \rangle}{\langle v_1, v_1 \rangle} v_1 - \frac{\langle u_3, v_2 \rangle}{\langle v_2, v_2 \rangle} v_2 \),
etc. Then \( \{v_i\} \) is orthogonal, and \( \operatorname{span}\{v_1,\dots,v_k\} = \operatorname{span}\{u_1,\dots,u_k\} \). Normalize to get orthonormal basis.
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Proof Idea: Each \( v_k \) is orthogonal to previous \( v_i \) by construction. Linear independence follows.
Adjoint & Special Operators
- Adjoint: For linear \( T: V \to V \), the adjoint \( T^* \) satisfies
$$ \langle T(x), y \rangle = \langle x, T^*(y) \rangle \quad \forall x,y \in V. $$
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Matrix Representation: If \( \mathcal{B} \) is orthonormal, then \( [T^*]_{\mathcal{B}} = [T]_{\mathcal{B}}^* \) (conjugate transpose).
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Self-Adjoint (Hermitian): \( T = T^* \).
Theorem: Eigenvalues of a Hermitian operator are real.
Proof: If \( T(v)=\lambda v \), then \( \langle T(v), v \rangle = \lambda \langle v,v \rangle \) and \( \langle v, T(v) \rangle = \overline{\lambda} \langle v,v \rangle \). Since \( \langle T(v), v \rangle = \langle v, T(v) \rangle \), we get \( \lambda = \overline{\lambda} \).
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Unitary: \( T^* T = TT^* = I \) (preserves inner product). Eigenvalues have absolute value 1.
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Normal: \( T T^* = T^* T \).
Spectral Theorem: A linear operator on a finite-dimensional complex inner product space is unitarily diagonalizable iff it is normal.
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[!TIP]
For real matrices, self-adjoint = symmetric (\( A^T=A \)), unitary = orthogonal (\( A^T A = I \)).
Orthogonal Projections & Open Subspaces
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Orthogonal Complement: \( W^\perp = \{ x \in V \mid \langle x, w \rangle = 0 \; \forall w \in W \} \). Then \( V = W \oplus W^\perp \) (direct sum).
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Characterization of Best Approximation: \( x \in W^\perp \) iff \( \|x - y\| \geq \|x\| \) for all \( y \in W \).
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Open Subspaces Theorem: In an inner product space (with norm topology), the only open subspace is \( V \) itself.
Proof: If \( W \) is open and proper, then there exists \( v \notin W \). Since \( W \) is a subspace, for any \( w \in W \), \( \|w\| \) can be arbitrarily small (scale \( w \)), so small balls around 0 are in \( W \). But then any vector \( u \) can be written as \( u = \frac{\varepsilon u}{\|u\|} \cdot \frac{\|u\|}{\varepsilon} \) for small \( \varepsilon \), and since \( \frac{\varepsilon u}{\|u\|} \) has small norm, it is in \( W \), so \( u \in W \) by subspace property. Contradiction.
VII. BILINEAR & QUADRATIC FORMS
Bilinear Forms
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Definition: A bilinear form on \( V \) is a map \( B: V \times V \to F \) linear in each argument.
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Matrix Representation: Given basis \( \mathcal{B} \), there exists unique matrix \( A \) such that
$$ B(x,y) = [x]_{\mathcal{B}}^T A [y]_{\mathcal{B}}. $$
Rank of \( B \) = rank of \( A \).
- Rank 1 Characterization:
$$ \text{rank}(B)=1 \iff B(x,y) = f(x) g(y) \text{ for some linear functionals } f,g. $$
- Proof: If \( B(x,y)=f(x)g(y) \), then matrix is \( u v^T \) where \( u, v \) are coordinate vectors of \( f,g \), so rank ≤1 and nonzero ⇒ rank 1. Conversely, if rank 1, matrix \( A = u v^T \), so \( B(x,y) = (x^T u)(v^T y) = f(x)g(y) \) with \( f(x)=x^T u \), \( g(y)=v^T y \).
Symmetric & Skew-Symmetric Bilinear Forms
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Symmetric: \( B(x,y) = B(y,x) \) for all \( x,y \). Matrix \( A \) is symmetric (\( A^T = A \)).
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Skew-Symmetric (Alternating in char ≠2): \( B(x,y) = -B(y,x) \). Matrix \( A \) is skew-symmetric (\( A^T = -A \)). Also, \( B(x,x)=0 \) for all \( x \).
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Example on \( \mathbb{R}^n \): Basis for skew-symmetric forms: \( \{ e_i \wedge e_j \mid 1 \leq i < j \leq n \} \), where \( (e_i \wedge e_j)(x,y) = x_i y_j - x_j y_i \). Dimension \( \binom{n}{2} \).
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[!TIP]
Over \( \mathbb{R} \), a symmetric bilinear form is positive definite iff all eigenvalues of its matrix are positive.
Group-Preserving Bilinear Forms
- Definition: A bilinear form \( B \) on \( V \) is preserved by a group \( G \subseteq GL(V) \) if
$$ B(gx, gy) = B(x,y) \quad \forall g \in G, \; \forall x,y \in V. $$
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Example: Standard dot product on \( \mathbb{R}^n \) is preserved by \( O(n) = \{ A \in GL_n(\mathbb{R}) \mid A^T A = I \} \).
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Invariant Forms under \( O(n,\mathbb{C}) \): On \( \mathbb{C}^n \), the only bilinear forms preserved by \( O(n,\mathbb{C}) \) are scalar multiples of the standard dot product \( B(x,y) = x^T y \).
Reason: If \( B(x,y) = x^T M y \) and \( A^T A = I \) for all \( A \in O(n,\mathbb{C}) \), then \( A^T M A = M \) for all such \( A \). This implies \( M \) is scalar (since \( O(n,\mathbb{C}) \) acts irreducibly on \( \mathbb{C}^n \) for \( n \geq 2 \)).
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Example (Jun 2024): On \( M_n(\mathbb{R}) \), \( B(A,B) = \operatorname{tr}(A^T B) \). For \( Q \in O(n) \),
\( B(QA, QB) = \operatorname{tr}((QA)^T (QB)) = \operatorname{tr}(A^T Q^T Q B) = \operatorname{tr}(A^T B) = B(A,B) \). So preserved.
VIII. SPECIAL TOPICS & PROOFS
Nilpotent Operators
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Definition: \( T \) is nilpotent if \( T^k = 0 \) for some \( k \geq 1 \).
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Examples:
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Differentiation operator \( D \) on \( P_n \) (polynomials of degree ≤ \( n \)): \( D^{n+1}=0 \).
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Strictly upper triangular matrices.
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Theorem: If \( A \) is nilpotent, then the commutator operator \( T(B) = AB - BA \) on \( M_n \) is nilpotent.
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Similarity Criterion: Two nilpotent operators are similar iff they have the same invariant factors (equivalently, same Jordan block structure).
Triangularizability
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Theorem: If all eigenvalues of \( T \) lie in \( F \), then there exists a basis of \( V \) such that \( [T] \) is upper triangular.
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Proof Idea: Use induction on \( \dim V \). Pick eigenvalue \( \lambda \), eigenvector \( v \). Extend \( \{v\} \) to a basis. Since \( T(v) = \lambda v \), the matrix has first column \( [\lambda, 0, \dots, 0]^T \). By induction on quotient \( V/\langle v \rangle \), get triangular form.
Minimal Polynomial of a Restriction
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Theorem: If \( W \) is \( T \)-invariant, then the minimal polynomial of \( T|_W \) divides the minimal polynomial of \( T \).
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Proof: Let \( p(t) \) be min poly of \( T \). Then \( p(T)=0 \). Restrict to \( W \): \( p(T|_W)=0 \). So min poly of \( T|_W \) divides \( p(t) \).
IX. SHORT NOTE TOPICS (FREQUENTLY ASKED)
Direct Sum Decompositions
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Internal vs External: \( V = U_1 \oplus \cdots \oplus U_k \) (internal) means each \( U_i \subseteq V \) and sum is direct. External direct sum \( U_1 \oplus \cdots \oplus U_k \) is the set of tuples with componentwise operations.
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Relation to Invariant Subspaces: If \( V = W_1 \oplus W_2 \) with \( W_i \) \( T \)-invariant, then \( [T] \) is block diagonal. Conversely, block diagonal matrix gives direct sum decomposition into invariant subspaces.
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Primary Decomposition: A special case where \( W_i = \ker p_i(T)^{e_i} \) from distinct irreducible factors of min poly.
Unitary and Normal Linear Transformations
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Unitary: \( T^* T = I \). Preserves inner product: \( \langle T(x), T(y) \rangle = \langle x, y \rangle \). Eigenvalues lie on unit circle.
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Normal: \( T T^* = T^* T \). Includes self-adjoint, unitary, and orthogonal (real case) operators.
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Spectral Theorem: On finite-dimensional complex inner product space, \( T \) is normal iff it is unitarily diagonalizable. On real spaces, symmetric operators are orthogonally diagonalizable.
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Consequences: Normal operators have orthogonal eigenspaces for distinct eigenvalues.
Annihilator of a Subspace
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Definition: \( W^0 = \{ f \in V^* \mid f(w)=0 \; \forall w \in W \} \).
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Properties:
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Subspace of \( V^* \).
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\( \dim W + \dim W^0 = \dim V \).
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\( (W^0)^0 = W \) (double annihilator).
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If \( W_1 \subseteq W_2 \), then \( W_2^0 \subseteq W_1^0 \).
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Use: Annihilators describe dual relationships; e.g., \( (U+W)^0 = U^0 \cap W^0 \).
Dual Spaces
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Definition: \( V^* = \mathcal{L}(V, F) \).
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Dual Basis: Given basis \( \{v_i\} \) of \( V \), there is unique dual basis \( \{f_i\} \) with \( f_i(v_j)=\delta_{ij} \). Then any \( f \in V^* \) is \( f = \sum f(v_i) f_i \).
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Double Dual: Natural isomorphism \( V \cong V^{**} \) via \( v \mapsto \hat{v} \) where \( \hat{v}(f)=f(v) \). This isomorphism is canonical (basis-free).
Primary Decomposition Theorem
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Statement: As in Section III.
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Significance: Reduces study of \( T \) to study on primary components \( \ker p_i(T)^{e_i} \), where the minimal polynomial is a power of an irreducible. Useful for computing Jordan form and functions of \( T \).
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Computation: From min poly \( p_T = \prod p_i^{e_i} \), each \( W_i = \ker p_i(T)^{e_i} \) is invariant, and \( V = \bigoplus W_i \).
Invariant Subspaces
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Definition: \( W \) is \( T \)-invariant if \( T(W) \subseteq W \).
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Examples:
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Eigenspaces.
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\( \ker T \), \( \operatorname{im} T \).
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For diagonal \( T \), any coordinate subspace.
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For Jordan block, the span of first \( k \) basis vectors.
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Finding Invariant Subspaces: Use polynomials in \( T \): \( \ker p(T) \), \( \operatorname{im} p(T) \) are invariant. Also, sums and intersections of invariant subspaces are invariant.
Jordan Blocks
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Definition: \( J(\lambda,k) \) as above.
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Role in JCF: Every matrix (over algebraically closed field) is similar to a direct sum of Jordan blocks. The block structure encodes:
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Number of blocks for \( \lambda \) = geometric multiplicity of \( \lambda \).
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Sizes of blocks determined by dimensions of \( \ker(A-\lambda I)^k \).
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Example: For eigenvalue \( \lambda \) with AM=4, GM=2, possible block structures: two blocks of size 2, or one of size 3 and one of size 1.
Cayley-Hamilton Theorem
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Statement: \( p_A(A)=0 \).
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Proof Sketch: Use adjugate matrix: \( (tI-A) \operatorname{adj}(tI-A) = p_A(t) I \). Substitute \( t=A \) to get \( 0 = p_A(A) I \).
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Applications:
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Compute \( A^n \) for \( n \geq \dim \).
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Find \( A^{-1} \) if invertible.
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Show that \( A \) satisfies polynomials dividing \( p_A \).
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Bilinear Forms
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Definition: \( B: V \times V \to F \) linear in each argument.
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Matrix Representation: \( B(x,y) = x^T A y \) after choosing basis.
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Rank: \( \operatorname{rank}(B) = \operatorname{rank}(A) \).
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Types: Symmetric (\( A^T=A \)), skew-symmetric (\( A^T=-A \)), alternating (\( B(x,x)=0 \)).
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Change of Basis: If \( P \) is change-of-basis matrix, new matrix is \( P^T A P \).
Inner Product Spaces
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Definition: As in Section VI.
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Key Properties:
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Cauchy-Schwarz: \( |\langle x,y \rangle| \leq \|x\| \|y\| \).
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Triangle inequality: \( \|x+y\| \leq \|x\| + \|y\| \).
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Pythagorean: If \( \langle x,y \rangle =0 \), then \( \|x+y\|^2 = \|x\|^2 + \|y\|^2 \).
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Examples: \( \mathbb{R}^n \) (dot), \( \mathbb{C}^n \) (standard), \( C[a,b] \) (integral), \( M_n \) with \( \langle A,B \rangle = \operatorname{tr}(A^T B) \).
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Orthonormal Basis: Exists via Gram-Schmidt. Coordinates: \( x = \sum \langle x, e_i \rangle e_i \).
Self-Adjoint (Hermitian) Operators
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Definition: \( T = T^* \). For matrices: \( A = A^* \) (conjugate transpose).
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Properties:
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Eigenvalues are real.
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Eigenvectors for distinct eigenvalues are orthogonal.
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Unitarily diagonalizable (spectral theorem).
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Example: Real symmetric matrices are self-adjoint w.r.t. dot product.
Symmetric/Skew-Symmetric Bilinear Forms
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Symmetric: \( B(x,y)=B(y,x) \). Matrix symmetric. Over \( \mathbb{R} \), can diagonalize by orthogonal transformation (spectral theorem for symmetric forms).
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Skew-Symmetric: \( B(x,y)=-B(y,x) \). Matrix skew-symmetric. Over \( \mathbb{R} \), can write \( B(x,y) = x^T A y \) with \( A^T=-A \). Standard example: \( x_1 y_2 - x_2 y_1 \) on \( \mathbb{R}^2 \). Basis: \( e_i \wedge e_j \).
Group-Preserving Bilinear Forms
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Definition: \( B(gx, gy)=B(x,y) \) for all \( g \in G \subseteq GL(V) \).
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Example: \( O(n) \) preserves standard dot product on \( \mathbb{R}^n \).
Verification: For \( Q \in O(n) \), \( (Qx) \cdot (Qy) = x^T Q^T Q y = x^T y \).
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Invariant Forms under \( O(n,\mathbb{C}) \): On \( \mathbb{C}^n \), only scalar multiples of \( x^T y \).
On \( M_n(\mathbb{C}) \), \( B(A,B)=\operatorname{tr}(A^T B) \) is preserved by \( O(n,\mathbb{C}) \) acting by left multiplication.