Matrix Transformation Examples and Practice Problems

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of A
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$ 
Find a vector in the null space of A that is not the zero vector 
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Give the elementary matrix that converts
$\left[\begin{array}{ccc}-2& -2& -1\\ -3& -1& -3\\ 1& -4& -3\end{array}\right]$

to

$\left[\begin{array}{ccc}-6& -2& -1\\ -5& -1& -3\\ -7& -4& -3\end{array}\right]$

Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of ${30}^{\circ }$. (b) Find the standard matrix of $T,[T]$. See p. 216 and, more generally, section 3.6 of your text if you're unsure of what this is.

Proove that the set of oil $2\times 2$ matrices with entries from R and determinant +1 is a group under multiplication

(7) If A and B are a square matrix of the same order. Prove that $(AB{A}^{-1}{)}^3=A{B}^3{A}^{-1}$

If $A=[1,2,4,3]$, find B such that $A+B=0$

Write the given matrix equation as a system of linear equations without matrices.
${\displaystyle \left[\begin{array}{ccc}2& 0& -1\\ 0& 3& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6\\ 9\\ 5\end{array}\right]}$

Suppose A is a $6\times 9$ matrix. If Nullity ${\displaystyle \left(AT{A}^{T}AT\right)}$ $=2$ then Nullity $(A)=2$

Write an augumented matrix for the system of linear equations
${\displaystyle \left[\begin{array}{c}x-y+z=8\\ y-12z=-15\\ z=1\end{array}\right]}$

1:Find the determinant of the following mattrix ${\displaystyle \left[\begin{array}{c}\begin{array}{ccc}2& -1& -6\end{array}\\ \begin{array}{ccc}-3& 0& 5\end{array}\\ \begin{array}{ccc}4& 3& 0\end{array}\end{array}\right]}$

2: If told that matrix A is singular Matrix find the possible value(s) for x ${\displaystyle A=\{\begin{array}{cc}16x& 4x\\ x& 9\end{array}}$

Demonstrate that W is the collection of all $3\times 3$ upper triangular matrices.
forms a subspace of all $3\times 3$ matrices. 
What is the dimension of W? Find a basis for W.

Let D be the diagonal subset ${\displaystyle D=\{(x,x)\mid x\in S_3\}}$ of the direct product $S_3\times S_3$. Prove that D is a subgroup of $S_3\times S_3$ but not a normal subgroup.

Find basis and dimension ${\displaystyle \{x\in R^4\mid xA=0\}}$ where ${\displaystyle A={[-1,1,2,1,1,0,2,3]}^T}$

Let ${\displaystyle u=i+2j-3k}$ and ${\displaystyle v=2i+3j+k\in R^3}$
(a) What is $u\cdot v$?
(b) What is $u\cdot v$?

Solve the following pair of linear equations by the elimination method and the substitution method:
$x+y=5,2x-3y=4$