Answer to Question #101023 in Linear Algebra for Alexander San Jose

Question

1. Let Upper T Subscript Upper A Baseline colon Upper R Superscript 3 Baseline right-arrow Upper R Superscript 3 be multiplication by A and find the dimension of the subspace of Upper R Superscript 3 consisting of all vectors x for which Upper T Subscript Upper A Baseline left-parenthesis x right-parenthesis equals 0.

a. A= 1 1 0
1 0 3
1 0 3
The dimension is????

b. A = 1 4 0
1 4 0
1 4 0
The dimension is???

c. A = 3 0 0
-3 3 0
3 3 3
The dimension is????

Answer 1

$Ax=0, \quad x= \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$

$a) A = \begin{bmatrix} 1&1&0 \\ 1&0&3 \\ 1&0&3 \end{bmatrix}$

$Ax=0 \; \Leftrightarrow \begin{cases} x_1+x_2=0 \\ x_1+3x_3=0 \\ x_1+3x_3=0 \end{cases}$

$x_1=-x_2=-3x_3$

$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -3x_3 \\ 3x_3 \\ x_3 \end{bmatrix} = x_3\begin{bmatrix} -3 \\ 3 \\ 1 \end{bmatrix}, \forall x_3 \in \mathbb{R}$

$\Rightarrow dimension \; of \; sub space \; of \; \mathbb{R}^3 \; consisting \; of \; all \; vectors \; x \; for \; which \; Ax=0$

$equals \; 1.$

$b) A = \begin{bmatrix} 1&4&0 \\ 1&4&0 \\ 1&4&0 \end{bmatrix}$

$Ax=0 \; \Leftrightarrow \begin{cases} x_1+4x_2=0 \\ x_1+4x_2=0 \\ x_1+4x_2=0 \end{cases}$

$x_1=-4x_2$

$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -4x_2 \\ x_2 \\ x_3 \end{bmatrix} = x_2\begin{bmatrix} -4 \\ 1 \\ 0 \end{bmatrix}+x_3 \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \forall x_2,x_3 \in \mathbb{R}$

$\Rightarrow dimension \; of \; sub space \; of \; \mathbb{R}^3 \; consisting \; of \; all \; vectors \; x \; for \; which \; Ax=0$

$equals \; 2.$

$c) A = \begin{bmatrix} 3&0&0 \\ -3&3&0 \\ 3&3&3 \end{bmatrix}$

$Ax=0 \; \Leftrightarrow \begin{cases} x_1=0 \\ -3x_1+3x_2=0 \\ 3x_1+3x_2+3x_3=0 \end{cases}$

$x_2=x_1=0, x_3=-x_1-x_2=0$

$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

$\Rightarrow dimension \; of \; sub space \; of \; \mathbb{R}^3 \; consisting \; of \; all \; vectors \; x \; for \; which \; Ax=0$

$equals \; 0.$