Prove the difference of any even integer minus any odd integer is odd.

The solution of initial value problem $y^{″}+2y^{\prime }+2y=\delta (t-\pi ),y(0)=0,y^{\prime }(0)=1$ and draw the graph of the solution.

Whether the statement “ Let A be an $m\times n$ matrix.
The system $Ax=b\text{is consistent for all b in}{R}^m\text{if and only if the columns of A form a generating set for}{R}^m$ " is true or false.

Describe how to obtain the graph of g and f if $g(x)=2(x+2{)}^2-3,h(x)=2x^2,andh(x)=2(x-3{)}^2+2$

(a) identify the transformation, and
(b) graphically represent the transformation for an arbitrary vector in ${\displaystyle R^2}$.
T(x, y) = (x, 2y)

Creating a transformation matrix
This question is related to this one, but more specific. I was given the following question:
Let ${\displaystyle D=\{d_1,d_2\}\text{and}B=\{b_1,b_2\}}$ be bases for vector spaces V and W respectively. Let ${\displaystyle T:V\to W}$ be a linear transformation with the property that ${\displaystyle T\left(d_1\right)=-5b_1-7b_2\text{and}T\left(d_2\right)=9b_1+4b_2}$. Find the matrix for T relative to D and B.
This is a pretty simple question and I just took the transformation properties and made a matrix:
$(\begin{array}{cc}-5& -7\\ 9& 4\end{array})$
The answer key has the same answer but with the rows and columns switched:
$(\begin{array}{cc}-5& 9\\ -7& 4\end{array})$
Are these both correct?

Properties of dual linear transformation
Let V,W finite dimensional vector spaces over the field F, and let ${\displaystyle T:V\to W}$ a linear transformation.
Define ${\displaystyle T\cdot :W\cdot \to V\cdot }$ the dual linear transformation. i.e ${\displaystyle T\cdot \left(\psi \right)=\psi \circ T}$.

To calculate:
The reduced form of the provided matrix, $\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$ with the use of row operations.