24-25-1-线性代数-期末

Question 1

(True or False) if $\mathbf{A}$ is a $4\times3$ matrix with rank 3, then $\mathbf{A}x=b$ is consistent. 【暂无答案】
(True or False) The set of all $n\times n$ real upper triangular matrices with usual matrix addition and scalar multiplication forms a vector space. 【暂无答案】
(True or False) If $\mathbf{A}$ is an $m\times n$ matrix with independent columns then the matrix $\mathbf{A}^T\mathbf{A}$ is nonsingular. 【暂无答案】
If a matrix $\mathbf{A}$ satisfies $\mathbf{A}^2-\mathbf{A}+4\mathbf{I}=\mathbf{O}$ , then $(\mathbf{A}+\mathbf{I})^{-1} =$ 【暂无答案】.
Let $\mathbf{A}$ be an $n\times n$ matrix with $\det(\mathbf{A})=2$ , then $\det(-3\mathbf{A}^{-1}) =$ 【暂无答案】.
Let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be vectors in $\mathbb{R}^4$ , then

$\beta_1=\alpha_1+\alpha_2$ ,

$\beta_2=\alpha_2+\alpha_3$ ,

$\beta_3=\alpha_3+\alpha_4$ ,

$\beta_4=\alpha_4+\alpha_1$ ,

are 【暂无答案】 (linearly independent / dependent).

The general form equation of the plane that is perpendicular to $(3,1,-1)^T$ and that passes through the point $(1,1,1)$ is 【暂无答案】.
The direction vector of the line

\begin{cases} x+y+z=1\\ 2x+z=2 \end{cases}

is 【暂无答案】.

Suppose that the $3\times3$ matrix $\mathbf{A}$ has eigenvalues $1,2,3$ . The trace of $\mathbf{A}+\mathbf{I}$ is 【暂无答案】
The quadratic form $x^2+4xy+2y^2$ is 【暂无答案】 (positive-definite, negative-definite).

Evaluate the determinant

\mathbf{D}=\begin{bmatrix} 1&b&0&0\\ a&1&b&0\\ 0&a&1&b\\ 0&0&a&1 \end{bmatrix}

where $a,b$ are scalars.

Suppose that $\mathbf{A}\mathbf{X}=\mathbf{B}-3\mathbf{X}$ , where

\mathbf{A}=\begin{bmatrix}-2&1&1\\1&-1&2\\1&2&0\end{bmatrix},\ \mathbf{B}=\begin{bmatrix}1&4\\1&2\\3&5\end{bmatrix}

Find $\mathbf{X}$ .

Consider the linear system

\begin{cases} x_1+x_2+x_3=4\\ a x_2+x_3=2\\ x_1+2x_2+2x_3=b \end{cases}

(a) For what values of $a$ and $b$ is the system inconsistent?

(b) For what values of $a$ and $b$ is the system consistent with infinitely many solutions? Find all the solutions.

Let

\mathbf{A} = (\alpha_1, \alpha_2, \alpha_3, \alpha_4) = \begin{pmatrix} 1 & 5 & -4 & -10 \\ 2 & 3 & -1 & -6 \\ 0 & -1 & 7 & 2 \\ 1 & 6 & -5 & 1 \end{pmatrix}

(a) Determine the rank of $\mathbf{A}$ and the dimension of the null space $N(\mathbf{A})$ of $\mathbf{A}$ .

(b) Find a basis for the column space of $\mathbf{A}$ .

Let $L:\mathbb{R}^3\to\mathbb{R}^2$ be the linear transformation defined by

L\left((x_1, x_2, x_3)^T\right)=\left(x_1+2x_2+x_3,\ 3x_1+6x_2+3x_3\right)^T

(a) Find the standard matrix representation of $L$ .

(b) Find the kernel and the range of $L$ .