Linear Algebra Practice Set 3

Let $x, a, b$ be real constants satisfying $a \neq b$. Let $A = (a_{ij})_{n \times n}$ be an $n$-th order matrix $(n \geq 2)$ satisfying:

$a_{kk} = x$ for all $k = 1, \ldots, n$

$a_{pq} = a$ for all $p, q = 1, \ldots, n$ with $p < q$

$a_{lm} = b$ for all $l, m = 1, \ldots, n$ with $l > m$

Calculate the determinant $|A|$.

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

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

Let $L = L(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ be the subspace generated by these vectors.

(1) Prove that basis (I): $\alpha_1, \alpha_2$ and basis (II): $\alpha_3, \alpha_4$ are both bases 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) Let $M = \lim_{k \to +\infty} M_k$. Prove that $M$ is idempotent, i.e., $M^2 = M$.

For the function space $V$ of $x \in (0, 1)$, define addition and scalar multiplication as:

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

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

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

(3) Transform $f(x) = x$ and $g(x) = x + e^x$ into an orthonormal basis $\eta_1, \eta_2$ of the Euclidean space with inner product $(f, g) = \int_0^1 f(x)g(x)\,dx$.

Given the quadratic form:

$f(x_1, x_2, x_3) = x_1x_2 - x_1x_3$

Find the canonical form of this quadratic form, its rank, and signature.

For an $m \times n$ matrix $A$ with $r \leq \min(m, n)$, prove that:

$r(A) = r$ if and only if $A$ can be decomposed as $A = \sum_{i=1}^{r} \alpha_i\beta_i^T$, where $\alpha_1, \ldots, \alpha_r$ and $\beta_1, \ldots, \beta_r$ are linearly independent column vectors.

(1) Let $A$ be an $n$-th order real symmetric matrix and $B$ be an $n$-th order positive definite matrix. Prove that the eigenvalues of $AB$ are all real;

(2) Let $A, B$ be real matrices satisfying $A^{-1} + B^{-1} = (A + B)^{-1}$. Prove that $(AB^{-1})^3 = E$.