Linear Algebra Practice Set 1

Let $A$ be an $n$-th order matrix satisfying $A^2 - A - 2E = O$, where $E$ is the identity matrix. Prove that $A$ is invertible and find $A^{-1}$.

Let $\mathbb{R}^4$ have two bases:

(I): $\alpha_1 = (1, 1, 0, -1)^T, \alpha_2 = (1, 2, 1, 0)^T, \alpha_3 = (1, 0, -1, -2)^T, \alpha_4 = (2, 0, -1, -1)^T$

(II): $\beta_1 = (1, 0, -1, 0)^T, \beta_2 = (0, 1, 1, 0)^T, \beta_3 = (1, 0, 1, -2)^T, \beta_4 = (0, 0, 0, 1)^T$

(1) Find the transition matrix from basis (I) to basis (II);

(2) If vector $\eta = (2, 3, 0, -1)^T$, find its coordinates under basis (I).

Given the linear system with parameter $\lambda$:

$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ x_1 + 2x_2 + 2x_3 + \lambda x_4 = \lambda \\ x_1 + 2x_2 + (\lambda + 2)x_3 + 2x_4 = \lambda + 2 \end{cases}$

(1) For what values of $\lambda$ does the system have infinitely many solutions?

(2) When the system has infinitely many solutions, find the general solution.

Let $A$ be an $n$-th order matrix with $\text{rank}(A) = r < n$. Let $A^*$ be the adjugate matrix of $A$ satisfying $A_{11} \neq 0$. Let $\alpha$ be a non-zero $n$-dimensional column vector.

Prove that the non-homogeneous linear system $AX = \alpha$ has infinitely many solutions if and only if $\alpha$ is a solution to the homogeneous system $A^*X = 0$.

Let $V$ be an $n$-dimensional real linear space. Let (I): $\varepsilon_1, \ldots, \varepsilon_n$ and (II): $\eta_1, \ldots, \eta_n$ be two bases of $V$.

Define $W = \{\alpha \in V : \alpha \text{ has the same coordinates under both bases}\}$

(1) Prove that $W$ is a subspace of $V$;

(2) If the rank of vectors $\varepsilon_1 - \eta_1, \ldots, \varepsilon_n - \eta_n$ is $r$, find $\dim(W)$.

Given the quadratic form:

$f(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 5x_3^2 + 2tx_1x_2 - 2x_1x_3 + 4x_2x_3$

(1) Find the value of $t$ such that the rank of this quadratic form is 2;

(2) For the value of $t$ found in (1), find an invertible linear transformation that transforms the quadratic form into its canonical form.

Let $A$ and $B$ be $n$-th order matrices with $A$ invertible. Let $r$ be a positive integer satisfying $0 < r < n$.

Define $M = \begin{pmatrix} E_r & O \\ B & -E_{n-r} \end{pmatrix}$ where $B$ is $r \times (n-r)$ and $O$ is the zero matrix.

Prove that $M$ is diagonalizable.

Let $\beta$ be a non-zero vector in Euclidean space $V$. Let $\beta_1, \beta_2, \beta_3$ be three vectors in $V$ satisfying:

(1) $(\beta_i, \beta) > 0$ for $i = 1, 2, 3$

(2) $(\beta_i, \beta_j) < 0$ for $i, j = 1, 2, 3, i \neq j$

Prove that $\beta_1, \beta_2, \beta_3$ are linearly independent.