Introduction to Linear Algebra (CY-401) Unit 1 Notes

I. FOUNDATIONS: VECTOR SPACES AND LINEAR TRANSFORMATIONS

A. Vector Spaces

Definition: A vector space $V$ over a field $F$ is a set with operations $+$ and $\cdot$ satisfying closure, associativity, commutativity, additive identity and inverses, distributivity, and $$\displaystyle 1\cdot v = v $$.
Examples: $$\displaystyle \mathbb{R}^n $$, $C[a,b]$ (continuous functions), $$\displaystyle P_n $$ (polynomials of degree $\leq n$).
Subspace: $W \subseteq V$ is a subspace iff $W \neq \emptyset$, and for all $u,v \in W$, $\alpha \in F$, we have $u+v \in W$ and $\alpha u \in W$.

[!TIP] To show $W$ is a subspace, verify it contains the zero vector and is closed under addition and scalar multiplication.
Direct sum:
- Internal: $$\displaystyle V = U_1 \oplus \cdots \oplus U_k $$ if every $v \in V$ can be written uniquely as $$\displaystyle v = u_1 + \cdots + u_k $$ with $$\displaystyle u_i \in U_i $$. Equivalent to $$\displaystyle V = \sum U_i $$ and $$\displaystyle U_i \cap \sum_{j \neq i} U_j = \{0\} $$ for all $i$.
- External: $$\displaystyle U_1 \oplus \cdots \oplus U_k $$ consists of tuples $$\displaystyle (u_1,\ldots,u_k) $$ with $$\displaystyle u_i \in U_i $$.
Quotient space: $$\displaystyle V/W = \{v + W \mid v \in V\} $$ with operations $$\displaystyle (v+W)+(u+W)=(v+u)+W $$, $$\displaystyle \alpha(v+W)=(\alpha v)+W $$.
- Dimension formula: $$\displaystyle \boxed{\dim(V/W) = \dim V - \dim W} $$.
- If $V$ is finitely generated, then $V/W$ is finitely generated (since $\dim(V/W) \leq \dim V$).
Basis and dimension: A basis is a linearly independent spanning set. Dimension $\dim V$ is the number of vectors in any basis.

B. Linear Transformations

Definition: $T: V \to W$ is linear if $$\displaystyle T(\alpha u + \beta v) = \alpha T(u) + \beta T(v) $$ for all $u,v \in V$, $\alpha,\beta \in F$.
Verification: Check additivity and homogeneity separately.

[!EXAMPLE] For $$\displaystyle T: \mathbb{R}^3 \to \mathbb{R}^4 $$ defined by $$\displaystyle T(x,y,z) = (x-2y+z,\; 2x-y+2z,\; x+2y+z,\; 2x+y+z) $$, verify $$\displaystyle T(\alpha (x_1,y_1,z_1) + \beta (x_2,y_2,z_2)) = \alpha T(x_1,y_1,z_1) + \beta T(x_2,y_2,z_2) $$ component-wise.
Kernel and image:
- $$\displaystyle \ker(T) = \{v \in V \mid T(v) = 0\} $$.
- $$\displaystyle \operatorname{im}(T) = \{T(v) \mid v \in V\} $$.
Rank-nullity theorem:

$$\boxed{\dim V = \dim \ker(T) + \dim \operatorname{im}(T)}$$

Proof sketch: Extend a basis of $\ker(T)$ to a basis of $V$; the images of the added vectors form a basis of $\operatorname{im}(T)$.

Matrix representation: If $$\displaystyle B = \{v_1,\ldots,v_n\} $$ basis of $V$, $$\displaystyle B' = \{w_1,\ldots,w_m\} $$ basis of $W$, then $$\displaystyle [T]_{B,B'} $$ has $j$-th column as coordinates of $$\displaystyle T(v_j) $$ in $B'$.
Change of basis: If $P$ is change-of-basis matrix from $B$ to $B'$ (so $$\displaystyle [v]_{B'} = P^{-1}[v]_B $$), then $$\displaystyle [T]_{B'} = P^{-1} [T]_B P $$.

C. Dual Spaces and Annihilators

Dual space: $$\displaystyle V^* = \{ f: V \to F \mid f \text{ linear} \} $$.
Annihilator of $W \subseteq V$:

$$W^0 = \{ f \in V^* \mid f(w)=0 \text{ for all } w \in W \}.$$

Dimension formula:

$$\boxed{\dim W^0 = \dim V - \dim W}$$

Proof: Choose basis $$\displaystyle \{v_1,\ldots,v_k\} $$ for $W$, extend to basis $$\displaystyle \{v_1,\ldots,v_n\} $$ for $V$. Dual basis $$\displaystyle \{f_1,\ldots,f_n\} $$ for $$\displaystyle V^* $$. Then $$\displaystyle W^0 = \operatorname{span}\{f_{k+1},\ldots,f_n\} $$.

Double dual: $$\displaystyle V^{**} = (V^*)^* $$. Natural isomorphism $$\displaystyle \phi: V \to V^{**} $$ given by $$\displaystyle \phi(v)(f) = f(v) $$ is an isomorphism when $V$ finite-dimensional.

D. Advanced Linear Transformation Topics

Invariant subspace: $W \subseteq V$ is $T$-invariant if $T(W) \subseteq W$.

[!TIP] Eigenspaces and kernels of polynomials in $T$ are invariant.
Minimal polynomial: Monic polynomial $m(\lambda)$ of least degree such that $$\displaystyle m(T)=0 $$. Exists since $V$ finite-dimensional.
Cayley-Hamilton theorem: If $$\displaystyle p(\lambda) = \det(T - \lambda I) $$ is the characteristic polynomial, then $$\displaystyle p(T)=0 $$.

$$\boxed{p(T)=0}$$

Proof sketch: Over $F$, $T$ has an upper triangular matrix representation (if all eigenvalues in $F$). Then $p(T)$ is upper triangular with zeros on diagonal, hence zero matrix. By similarity, holds for any representation.

Primary decomposition theorem: If minimal polynomial factors as $$\displaystyle m(\lambda) = \prod_{i=1}^k p_i(\lambda)^{e_i} $$ with $$\displaystyle p_i $$ distinct irreducible, then

$$V = \bigoplus_{i=1}^k \ker(p_i(T)^{e_i}),$$

and each $$\displaystyle \ker(p_i(T)^{e_i}) $$ is $T$-invariant.

Nilpotent operator: $T$ is nilpotent if $$\displaystyle T^k=0 $$ for some $k$. Index is smallest such $k$.

[!EXAMPLE] Differentiation operator $D$ on $$\displaystyle P_n $$ (polynomials of degree $\leq n$) is nilpotent with index $n+1$ since $$\displaystyle D^{n+1}=0 $$ but $$\displaystyle D^n \neq 0 $$.

II. EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION

A. Characteristic Polynomial

Definition: For $T: V \to V$, $$\displaystyle p(\lambda) = \det(T - \lambda I) $$. For matrix $A$, $$\displaystyle p(\lambda) = \det(A - \lambda I) $$.
Eigenvalues: Roots of $$\displaystyle p(\lambda)=0 $$.
Algebraic multiplicity: Multiplicity of $\lambda$ as root of $p(\lambda)$.

B. Eigenvalues and Eigenvectors

Eigenvalue $\lambda$ and eigenvector $v \neq 0$: $$\displaystyle T(v) = \lambda v $$.
Eigenspace: $$\displaystyle E_\lambda = \{ v \mid T(v) = \lambda v \} = \ker(T - \lambda I) $$.
Geometric multiplicity: $$\displaystyle \dim E_\lambda $$.
Relationship: $\text{geo mult} \leq \text{alg mult}$.

C. Diagonalization

Diagonalizable: $\exists$ basis of $V$ consisting of eigenvectors of $T$. Equivalent to $$\displaystyle V = \bigoplus_\lambda E_\lambda $$.
Criteria: For each eigenvalue, $$\displaystyle \dim E_\lambda = $$ algebraic multiplicity, or $$\displaystyle \sum_\lambda \dim E_\lambda = \dim V $$.

[!TIP] Distinct eigenvalues $\Rightarrow$ diagonalizable.
Process:
1. Find eigenvalues from $$\displaystyle \det(A-\lambda I)=0 $$.
2. For each $\lambda$, find basis of $$\displaystyle E_\lambda $$.
3. Form $P$ with eigenvectors as columns; then $$\displaystyle P^{-1}AP = D $$ diagonal with eigenvalues.
[!EXAMPLE] Diagonalize $$\displaystyle A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \end{bmatrix} $$: eigenvalues $0,2,4$; eigenvectors $(1,-1,1)$, $(0,1,1)$, $(1,0,2)$; $$\displaystyle P = \begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \\ 1 & 1 & 2 \end{bmatrix} $$, $$\displaystyle D = \operatorname{diag}(0,2,4) $$.

D. Special Matrix Classes and Spectral Theorems

Hermitian (self-adjoint): $$\displaystyle A = A^* $$ (conjugate transpose). Eigenvalues are real; eigenvectors for distinct eigenvalues are orthogonal.
Unitary: $$\displaystyle A^* A = I $$. Eigenvalues lie on unit circle; unitarily diagonalizable.
Normal: $$\displaystyle A A^* = A^* A $$. Unitarily diagonalizable over $\mathbb{C}$.
$$\displaystyle A^* A $$: Eigenvalues are real and non-negative; $$\displaystyle A^* A $$ is unitarily diagonalizable.
Triangular form: If all characteristic roots lie in $F$, then $\exists$ basis where matrix of $T$ is upper triangular.

III. INNER PRODUCT SPACES

A. Definition and Properties

Inner product: $\langle \cdot, \cdot \rangle: V \times V \to F$ satisfying:
1. Linearity in first argument: $$\displaystyle \langle \alpha u + \beta v, w \rangle = \alpha \langle u,w \rangle + \beta \langle v,w \rangle $$.
2. Symmetry (conjugate symmetry over $\mathbb{C}$): $$\displaystyle \langle u,v \rangle = \overline{\langle v,u \rangle} $$.
3. Positive-definiteness: $$\displaystyle \langle v,v \rangle > 0 $$ for $v \neq 0$.
Examples:
- $$\displaystyle \mathbb{R}^n $$: $$\displaystyle \langle x,y \rangle = x^T y $$.
- $C[a,b]$: $$\displaystyle \langle f,g \rangle = \int_a^b f(x)\overline{g(x)}\,dx $$.
- $$\displaystyle P_n $$: $$\displaystyle \langle p,q \rangle = \int_0^1 p(x)q(x)\,dx $$ (over $\mathbb{R}$).
Norm: $$\displaystyle \|v\| = \sqrt{\langle v,v \rangle} $$.
Cauchy-Schwarz inequality:

$$\boxed{|\langle u,v \rangle| \leq \|u\| \|v\|}$$

Parallelogram law:

$$\boxed{\|u+v\|^2 + \|u-v\|^2 = 2\|u\|^2 + 2\|v\|^2}$$

B. Orthogonality and Orthogonal Complements

Orthogonal: $$\displaystyle \langle u,v \rangle = 0 $$.
Orthogonal set: $$\displaystyle \{v_i\} $$ with $$\displaystyle \langle v_i, v_j \rangle = 0 $$ for $i \neq j$.
Orthogonal complement: $$\displaystyle W^\perp = \{ v \in V \mid \langle v,w \rangle = 0 \text{ for all } w \in W \} $$.
Projection theorem: For finite-dimensional $V$, $$\displaystyle V = W \oplus W^\perp $$.
Property: $$\displaystyle (W^\perp)^\perp = W $$ for finite-dimensional $V$.

C. Adjoint and Special Operators

Adjoint $$\displaystyle T^* $$: Defined by $$\displaystyle \langle T(v), w \rangle = \langle v, T^*(w) \rangle $$ for all $v,w \in V$. Exists and unique in finite dimensions.
Self-adjoint (Hermitian): $$\displaystyle T = T^* $$.
Unitary: $$\displaystyle T^* = T^{-1} $$.
Normal: $$\displaystyle T T^* = T^* T $$.
Riesz representation theorem: For every $$\displaystyle f \in V^* $$, $\exists!$ $v \in V$ such that $$\displaystyle f(w) = \langle w, v \rangle $$ for all $w \in V$.

[!TIP] On $$\displaystyle \mathbb{R}^3 $$ with standard inner product, any linear functional $f$ satisfies $$\displaystyle f(\mathbf{r}) = \mathbf{r} \cdot \mathbf{a} $$ for some $$\displaystyle \mathbf{a} \in \mathbb{R}^3 $$.

D. Orthogonalization

Gram-Schmidt process: Given linearly independent $$\displaystyle \{v_1,\ldots,v_n\} $$, construct orthogonal $$\displaystyle \{u_1,\ldots,u_n\} $$:

$$ \begin{aligned} u_1 &= v_1, \\ u_2 &= v_2 - \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1, \\ u_k &= v_k - \sum_{i=1}^{k-1} \frac{\langle v_k, u_i \rangle}{\langle u_i, u_i \rangle} u_i. \end{aligned} $$

Proof by induction: Each $$\displaystyle u_k $$ is orthogonal to all previous $$\displaystyle u_i $$.

Orthonormal basis: Normalize each $$\displaystyle u_i $$: $$\displaystyle e_i = u_i / \|u_i\| $$.

[!EXAMPLE] Apply to $\{(1,1,1), (1,0,1), (1,1,0)\}$ in $$\displaystyle \mathbb{R}^3 $$ with dot product.

E. Topological Aspects

Norm topology: Open sets are unions of open balls $$\displaystyle B(v,r) = \{ w \mid \|w-v\| < r \} $$.
No proper open subspaces: If $U \subseteq V$ is a subspace with non-empty interior, then $$\displaystyle U = V $$.

Proof: Suppose $U$ proper subspace. If $U$ has interior, $\exists x \in U$, $$\displaystyle r>0 $$ such that $B(x,r) \subseteq U$. For any $y \in V$, let $$\displaystyle z = x + \frac{r}{2} \frac{y}{\|y\|} $$ (if $y \neq 0$). Then $$\displaystyle \|z-x\| = r/2 < r $$, so $z \in B(x,r) \subseteq U$. Since $U$ subspace, $z-x \in U$, so $$\displaystyle y = \frac{2\|y\|}{r} (z-x) \in U $$. Thus $V \subseteq U$, contradiction.

IV. BILINEAR FORMS

A. General Bilinear Forms

Definition: $B: V \times V \to F$ is bilinear if linear in each argument.
Matrix representation: For basis $$\displaystyle \{e_1,\ldots,e_n\} $$, $$\displaystyle B(v,w) = [v]^T A [w] $$, where $$\displaystyle A_{ij} = B(e_i, e_j) $$.
Rank: $$\displaystyle \operatorname{rank}(B) = \operatorname{rank}(A) $$.
Change of basis: If $P$ is change-of-basis matrix, new matrix is $$\displaystyle P^T A P $$.
Matrix equivalence: $A$ and $B$ are equivalent if $\exists$ invertible $P,Q$ such that $$\displaystyle B = P A Q $$.
Rank canonical form: Every matrix is equivalent to $$\displaystyle \operatorname{diag}(I_r, 0) $$, where $$\displaystyle r = \operatorname{rank}(A) $$.

[!TIP] Rank is invariant under equivalence.

B. Symmetric Bilinear Forms

Symmetric: $$\displaystyle B(v,w) = B(w,v) \iff A = A^T $$.
Diagonalization over $\mathbb{R}$: By spectral theorem, $\exists$ orthogonal $Q$ such that $$\displaystyle Q^T A Q = D $$ diagonal.
Sylvester’s law of inertia: For real symmetric forms, the numbers of positive, negative, and zero eigenvalues (inertia) are invariant under congruence.
Canonical form: $$\displaystyle \operatorname{diag}(\underbrace{1,\ldots,1}_{p}, \underbrace{-1,\ldots,-1}_{q}, \underbrace{0,\ldots,0}_{r}) $$, where $p,q,r$ are inertia indices.

C. Skew-Symmetric Bilinear Forms

Skew-symmetric: $$\displaystyle B(v,w) = -B(w,v) \iff A^T = -A $$.
Alternating (char $\neq 2$): $$\displaystyle B(v,v)=0 $$ for all $v$ $\iff$ skew-symmetric.
Canonical form over $\mathbb{R}$: $\exists$ basis where matrix is block diagonal with blocks $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$.
Basis for space of skew-symmetric forms on $$\displaystyle \mathbb{R}^n $$: Dimension $$\displaystyle \frac{n(n-1)}{2} $$. Standard basis: for $$\displaystyle 1 \leq i < j \leq n $$, define $$\displaystyle B_{ij} $$ by $$\displaystyle B_{ij}(e_i,e_j)=1 $$, $$\displaystyle B_{ij}(e_j,e_i)=-1 $$, others zero.

D. Group-Invariant Bilinear Forms

Invariant under group $G$: $$\displaystyle B(gv, gw) = B(v,w) $$ for all $g \in G$, $v,w \in V$.
For $O(n,\mathbb{R})$: Invariant forms are scalar multiples of the standard inner product.
Example on matrix space: On $$\displaystyle M_n(\mathbb{R}) $$, $$\displaystyle B(A,B) = \operatorname{tr}(A^T B) $$ is invariant under $O(n)$ (since $$\displaystyle Q^T A^T Q Q^T B Q = (QAQ^T)^T (QBQ^T) $$? Actually, $$\displaystyle B(QAQ^T, QBQ^T) = \operatorname{tr}((QAQ^T)^T QBQ^T) = \operatorname{tr}(A^T Q^T Q B Q^T Q) = \operatorname{tr}(A^T B) $$ if $Q$ orthogonal).

[!TIP] Invariant forms under a group are often multiples of trace forms or standard inner products.

V. CANONICAL FORMS FOR LINEAR OPERATORS

A. Jordan Canonical Form

Jordan block $J(\lambda, k)$: $\lambda$ on diagonal, $1$’s on superdiagonal, size $k \times k$.
Jordan form: Block diagonal matrix with Jordan blocks for each eigenvalue.
Existence and uniqueness: Over algebraically closed fields (e.g., $\mathbb{C}$), every operator has a unique Jordan form up to block ordering.
Determination:
- Characteristic polynomial gives eigenvalues and algebraic multiplicities.
- Minimal polynomial gives largest block size for each eigenvalue.
- For eigenvalue $\lambda$, the number of Jordan blocks of size $\geq s$ equals $$\displaystyle \dim \ker((T-\lambda I)^s) - \dim \ker((T-\lambda I)^{s-1}) $$.
Jordan basis: Chain of generalized eigenvectors: for block of size $k$, vectors $$\displaystyle v_1, \ldots, v_k $$ with $$\displaystyle (T-\lambda I)v_1=0 $$, $$\displaystyle (T-\lambda I)v_i = v_{i-1} $$ for $$\displaystyle i>1 $$.

[!EXAMPLE] Reduce $$\displaystyle A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix} $$:

Char poly: $$\displaystyle (\lambda-1)^2(\lambda-3) $$. For $$\displaystyle \lambda=1 $$, $$\displaystyle \dim \ker(A-I)=1 $$ (since $$\displaystyle A-I = \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{bmatrix} $$, rank 2, nullity 1), so one block of size 2. For $$\displaystyle \lambda=3 $$, one block of size 1. Jordan form: $$\displaystyle \operatorname{diag}\left( \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, [3] \right) $$.

Jordan basis: For $$\displaystyle \lambda=1 $$, take $$\displaystyle v_1 = (0,1,0)^T $$ (eigenvector), $$\displaystyle v_2 = (1,0,0)^T $$ with $$\displaystyle (A-I)v_2 = v_1 $$. For $$\displaystyle \lambda=3 $$, $$\displaystyle v_3 = (0,0,1)^T $$. Basis $$\displaystyle \{v_1, v_2, v_3\} $$.

B. Application to Nilpotent Operators

Nilpotent: All eigenvalues $0$; Jordan blocks all have $0$ on diagonal.
Similarity criterion: Two nilpotent operators are similar iff they have the same Jordan block sizes (i.e., same partition of the index).

[!TIP] Invariants: dimensions of $$\displaystyle \ker(T^k) $$ determine block sizes.

C. Rational Canonical Form (if referenced)

Companion matrix for monic polynomial $$\displaystyle p(\lambda)=\lambda^k + a_{k-1}\lambda^{k-1}+\cdots+a_0 $$:

$$ C(p) = \begin{bmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{k-1} \end{bmatrix}. $$

Invariant factors: Monic polynomials $$\displaystyle d_1 \mid d_2 \mid \cdots \mid d_t $$ such that $$\displaystyle V \cong \bigoplus_{i=1}^t F[\lambda]/(d_i) $$ as $F[\lambda]$-modules.
Rational canonical form: Block diagonal with companion matrices of invariant factors; unique up to ordering.

VI. SUPPLEMENTARY TOPICS AND PROOFS

A. Direct Sum Decompositions

Internal direct sum: $$\displaystyle V = U_1 \oplus \cdots \oplus U_k $$ iff $$\displaystyle V = \sum U_i $$ and each $$\displaystyle U_i \cap \sum_{j \neq i} U_j = \{0\} $$.
External direct sum: $$\displaystyle \bigoplus_{i=1}^k U_i = \{ (u_1,\ldots,u_k) \mid u_i \in U_i \} $$.
Primary decomposition: From minimal polynomial $$\displaystyle m = \prod p_i^{e_i} $$, $$\displaystyle V = \bigoplus \ker(p_i(T)^{e_i}) $$.

B. Short Note Topics (Frequent in Exams)

Cayley-Hamilton theorem: Every operator satisfies its characteristic polynomial. Proof: Use triangular form or polynomial division: write $$\displaystyle p(\lambda) = q(\lambda)(\lambda - \lambda_0) + r $$, then $$\displaystyle p(T)=q(T)(T-\lambda_0 I) + rI $$; show $$\displaystyle r=0 $$ by evaluating on eigenvectors.
Primary decomposition theorem: Decomposes $V$ into $T$-invariant subspaces corresponding to distinct irreducible factors of minimal polynomial. Proof sketch: Use that $$\displaystyle \ker(p_i(T)^{e_i}) $$ are invariant and their sum is direct and equals $V$.
Invariant subspaces: $W$ with $T(W) \subseteq W$. Examples: eigenspaces, kernels of polynomials in $T$. Key for diagonalization and Jordan form.
Dual space: $$\displaystyle V^* = \operatorname{Hom}(V,F) $$; $$\displaystyle \dim V^* = \dim V $$. Dual basis: if $$\displaystyle \{v_i\} $$ basis, $$\displaystyle \{f_i\} $$ with $$\displaystyle f_i(v_j)=\delta_{ij} $$.
Bilinear forms: General definition, matrix representation, rank, change of basis. Rank 1 forms: $$\displaystyle B(x,y)=f(x)g(y) $$ for linear functionals $f,g$.
Inner product spaces: Axioms, examples, Cauchy-Schwarz, parallelogram law, projection theorem.
Jordan blocks: Structure: $\lambda$ on diagonal, $1$’s on superdiagonal. Size determined by chain of generalized eigenvectors.
Self-adjoint operators: $$\displaystyle T=T^* $$. Spectral theorem: Over $\mathbb{R}$, symmetric matrices have real eigenvalues and orthogonal eigenvectors; over $\mathbb{C}$, Hermitian matrices unitarily diagonalizable.
Annihilator $$\displaystyle W^0 $$: $$\displaystyle \{ f \in V^* \mid f|_W = 0 \} $$; $$\displaystyle \dim W^0 = \dim V - \dim W $$.
Skew-symmetric bilinear forms: $$\displaystyle B(v,w)=-B(w,v) $$; canonical form with $2\times2$ blocks.
Unitary and normal transformations:
- Unitary: $$\displaystyle T^* = T^{-1} $$; preserves inner product.
- Normal: $$\displaystyle T T^* = T^* T $$; unitarily diagonalizable over $\mathbb{C}$.

[!IMPORTANT] Exam Focus: Past papers frequently test rank-nullity, Cayley-Hamilton, diagonalization criteria, Jordan form computation, Gram-Schmidt, and properties of Hermitian/unitary matrices. Always verify linearity, compute kernels/images carefully, and distinguish algebraic vs. geometric multiplicity.

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Introduction to Linear Algebra (CY-401) - Unit 1 Short Notes