Linear Algebra Practice Set 4

Given the determinant:

$D = \begin{vmatrix} a_1+x & a_2 & \cdots & a_n \\ a_1 & a_2+x & \cdots & a_n \\ \vdots & \vdots & \ddots & \vdots \\ a_1 & a_2 & \cdots & a_n+x \end{vmatrix}$

Prove that: $D = x^{n-1}\left(x + \sum_{i=1}^{n} a_i\right)$

Given vectors in $\mathbb{R}^3$:

$\alpha_1 = (2, 4, 2)^T, \alpha_2 = (1, 1, 0)^T, \alpha_3 = (2, 3, 1)^T, \alpha_4 = (3, 5, 2)^T$

These generate a subspace $L(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$.

(1) Prove that basis (I): $\alpha_1, \alpha_2$ is a basis of $L$; and basis (II): $\alpha_3, \alpha_4$ is also a basis of $L$;

(2) Find the transition matrix from basis (II) to basis (I).

Let $\lambda$ be a real parameter. Solve the linear system:

$\begin{cases} x_1 + x_2 - x_3 = 1 \\ 2x_1 + 3x_2 + \lambda x_3 = 3 \\ x_1 + \lambda x_2 + 3x_3 = 2 \end{cases}$

Let $A$ be a real matrix satisfying $6A^3 + 11A^2 - 6A + E = O$.

(1) Find the eigenvalues of $A$;

(2) Let $M_k = A^k = (a_{ij}^{(k)})$. Prove that $\lim_{k \to +\infty} a_{ij}^{(k)}$ exists;

(3) When $k \to +\infty$, let $M = \lim M_k$. Prove that $M$ is idempotent, i.e., $M^2 = M$.

For the function space $V$ of $x \in (0,1)$, define:

$f + g = f(x) + g(x), \quad kf = kf(x)$

(1) Prove that functions $f(x) = ax + be^x$ form a subspace;

(2) Prove that $f(x) = x, g(x) = x + e^x$ are linearly independent;

(3) Transform them into an orthonormal basis $\eta_1, \eta_2$ with inner product $(f,g) = \int_0^1 f(x)g(x)\,dx$.

Given quadratic form: $f(x_1,x_2,x_3) = x_1x_2 - x_1x_3$

Find the canonical form, rank, and signature.

For matrix $A_{m\times n}$ with $r(A) = r \leq \min(m,n)$, prove:

$r(A) = r \Leftrightarrow A = \sum_{i=1}^r \alpha_i\beta_i^T$

where $\alpha_1,\ldots,\alpha_r$ and $\beta_1,\ldots,\beta_r$ are linearly independent.

(1) Let $A$ be $n$-th order real symmetric, $B$ positive definite. Prove eigenvalues of $AB$ are real;

(2) If $A,B$ satisfy $A^{-1} + B^{-1} = (A+B)^{-1}$, prove $(AB^{-1})^3 = E$.