Expand using logarithmic properties. Where possible, evaluate logarithmic expressions.

Let T be a linear transformation from ${\displaystyle R^2\text{into}{R}^2}$ such that ${\displaystyle T(1,0)=(1,1)\text{and}T(0,1)=(-1,1)}$. Find ${\displaystyle T(6,1)\text{and}T(-5,1)}$
${\displaystyle T(6,1)=?}$
${\displaystyle T(-5,1)=?}$

To state:
The solution of the given initial value problem using the method of Laplace transforms.
Given:
The initial value problem is,
$y+2y^{\prime }+5y=0,y\left(0\right)=2,y^{\prime }\left(0\right)=4$

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 ${\displaystyle T\cdot \left(\psi \right)=\psi \circ T}$.

a) To find:
The images of the following points under under a ${90}^{\circ }$ rotation counterclockwise about the origin:
I. $(2,3)$
II. $(-1,2)$
III, (m,n) interms of m and n
b)To show:
That under a half-turn with the origin as center, the image of a point $(a,b)\text{has coordinates}(-a,-b).$
c) To find:
The image of $P(a,b)\text{under the rotation clockwise by}{90}^{\circ }$ about the origin.

To find: The solution of the given initial value problem.

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?

Guided Proof Let $v_1,v_2,....V_n$ be a basis for a vector space V.
Prove that if a linear transformation $T:V\to V$ satisfies
$T(v_i)=0fori=1,2,...,n,$ then T is the zero transformation.
To prove that T is the zero transformation, you need to show that $T(v)=0$ for every vector v in V.
(i) Let v be the arbitrary vector in V such that $v=c_1{v}_1+c_2{v}_2+\cdots +c_n{V}_n$
(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of $T(v_j)$ .
(iii) Use the fact that $T(v_j)=0$
to conclude that $T(v)=0,$ making T the zero transformation.