Introduction to Linear Algebra (CY-401) - Important Questions

Find the linear transformation / its value at a vector given its values on a basis.

Verify that a given mapping is a linear transformation.

Define subspace of a vector space and prove that the quotient of a finitely generated vector space is finitely generated.

State and prove the rank-nullity theorem.

Let $f$ be a linear functional on $R^3$. Show that there exists $\bar{a} \in R^3$ such that $f(\bar{r}) = \bar{r} \cdot \bar{a}$.

Determine whether a given concrete sum of subspaces of $F^3$ / $\mathbb{R}^3$ is a direct sum.

Prove that the canonical map $\eta: V \to V/W$ defined by $\eta(x)=x+W$ is a linear transformation.

Prove the modular law $U_1 \cap (U_2 + (U_1 \cap U_3)) = (U_1 \cap U_2) + (U_1 \cap U_3)$ for subspaces.

Prove every subspace has a complementary subspace with $V = V_0 \oplus V_1$.

Prove $\dim(V)=n \geq 1$ iff $V$ is the direct sum of $n$ one-dimensional subspaces.

Find two linear operators $T$ and $U$ on $R^2$ such that $TU = 0$ but $UT \neq 0$.

Prove the annihilator $W^0$ is a subspace of $V^*$ and find $\dim W^0$ in terms of $\dim V$ and $\dim W$.

Show that a given $3\times 3$ matrix / linear operator is diagonalizable and find the diagonal matrix and the transforming (modal) matrix $P$.

Prove that a linear operator on an $n$-dimensional vector space with $n$ distinct eigenvalues / characteristic values is diagonalizable.

Verify the Cayley-Hamilton theorem for a given $2\times 2$ matrix by finding its characteristic polynomial and substituting the matrix

Show that for $A = \begin{bmatrix} 1 & 1 & 0 & 0 \\ -1 & -1 & 0 & 0 \\ -2 & -2 & 2 & 1 \\ 1 & 1 & -1 & 0 \end{bmatrix}$ the characteristic polynomial is $x^2(x-1)^2$ and it is also the minimal polynomial.

State and prove the Cayley-Hamilton theorem

Write short note on any two:

Write short note on any two: Cayley-Hamilton theorem, primary decomposition theorem, invariant subspaces

Find the characteristic polynomial and minimal polynomial of the given $3\times 3$ matrix $B$.

Let $W$ be an invariant subspace for linear operator $T$. Prove that the minimal polynomial for the restriction operator $T_W$ divides the minimal polynomial for $T$, without referring to matrices.

Write a short note on characteristic values and characteristic vectors, and on algebraic multiplicity and geometric multiplicity.

Write a short note on Cayley-Hamilton theorem and annihilating polynomials

Prove that for the companion matrix $A = \begin{bmatrix} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a \end{bmatrix}$ the characteristic polynomial is $x^3-ax^2-bx-c$ and it is also the minimal polynomial.

Find the eigenvalues and eigenvectors of a $3\times 3$ symmetric matrix.

Show that a given vector space with a specified formula forms an inner product space by verifying the inner product axioms.

Define adjoint of Linear Transformation. The linear transformation $T:R^3 \to R^4$ defined by $T(x,y,z) = (x-2y+z, 2x+y+2z, 3x-y+z)$ then show that T is a linear transformation.

Let $W$ be a subspace of a finite-dimensional inner product space $V$; if $\langle x,y\rangle + \langle y,x\rangle \le \langle y,y\rangle$ for all $y \in W$, prove $x \in W^{\perp}$.

Show that there is no proper open subspace of an inner product space.

State and prove the Gram-Schmidt orthogonalization process.

State the essential axioms/properties of inner product spaces and give an example of a vector space satisfying them.

Prove $\alpha = \beta$ if and only if $(\alpha \mid \gamma) = (\beta \mid \gamma)$ for every $\gamma$ in an inner product space $V$.

Find the eigenvalues, corresponding eigenvectors, and the invariant subspaces for each distinct eigenvalue of a $2\times 2$ matrix.

Explain invariant subspace with suitable examples.

Find all eigenvalues and a basis for each eigenspace (eigenvectors) of a $2\times 2$ linear operator/matrix.

Prove the parallelogram law $\|\alpha+\beta\|^2 + \|\alpha-\beta\|^2 = 2\|\alpha\|^2 + 2\|\beta\|^2$ in an inner product space.

Find the Jordan canonical form of a given $3\times 3$ matrix and determine the corresponding Jordan basis.

Define Canonical form of linear transformation. Show that every $m \times n$ matrix is equivalent to unique matrix in one of canonical form.

Define Jordan canonical form and reduce a given matrix to Jordan canonical form.

Find the Jordan canonical form of a given $3\times 3$ matrix.

Prove that the commutator operator $T(B)=AB-BA$ on $n \times n$ matrices is nilpotent when $A$ is nilpotent.

Prove that the differentiation operator on the space of polynomials of degree at most $n$ is nilpotent.

Determine the possible Jordan canonical forms of a matrix from its characteristic polynomial.

Determine the primary decomposition of the vector space with respect to a given linear operator/matrix.

If $T \in A(V)$ has all it's characteristic roots in F, then there exists a basis of V such that matrix representation of T is triangular. Prove it.

Prove that two nilpotent linear transformations are similar if and only if they have the same invariants.

Explain Jordan blocks with suitable examples.

State and prove the primary decomposition theorem.

Find a basis and general form for the space of all skew-symmetric bilinear forms on $R^n$ (in particular $R^3$).

Find all bilinear forms on the space of $n\times 1$ column vectors invariant under the orthogonal group $O(n)$.

Write short notes on

Define characteristic polynomial and prove all eigenvalues of $A^*A$ are real and $A^*A$ is unitarily diagonalizable.

Prove that all eigenvalues of a Hermitian matrix are real.

Show that a bilinear form on $V$ is a product of linear functionals iff it has rank $1$.

Determine whether a given $3\times 3$ complex matrix operator is Hermitian.

For $B(A,B)=\mathrm{tr}(A^T B)$ on $O(n)$, verify whether $B$ preserves the group structure.

Determine whether a given $2\times 2$ complex matrix operator is unitary.

Write a detailed explanatory note on skew-symmetric bilinear forms.

Check whether a given linear operator on $C^2$ is self-adjoint and whether it is unitary.

Show that $f(x,y)=x^T A y$ for an $n\times n$ matrix $A$ defines a bilinear form.