Matrix Transformation Examples and Practice Problems

If the matrix of a linear transformation $:<!--\; :\; -->{\mathbb{R}}^N\to <!--\; \to \; -->{\mathbb{R}}^N$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to geometrically interpret the transformation in a nice/simple way?

Give the standard matrix of the linear transformation that first sends (x, y, z) to (y, y, z), and rotates this vector 90 degrees counterclockwise about the origin in the x = y plane. Find standard matrix of linear transformation.

Given a matrix $Y\in <!--\; \in \; -->{\mathbb{R}}^{m\times <!--\; \times \; -->n}$. Find a transformation matrix $\mathrm{\Theta <!--\; \Theta \; -->}\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->p}$ such that
$${\displaystyle \frac{1}{m}}{\mathrm{\Theta <!--\; \Theta \; -->}}^T{Y}^{T}Y\mathrm{\Theta <!--\; \Theta \; -->}=I_{p\times <!--\; \times \; -->p},$$
where 𝐼𝑝×𝑝 is identity matrix.
My attempt: ${\displaystyle \frac{1}{\sqrt{m}}}Y\mathrm{\Theta <!--\; \Theta \; -->}$ is orthogonal matrix and tried to find $\mathrm{\Theta <!--\; \Theta \; -->}$ satisfies it but that doesn't work.

Vector, $u:=[u_1,\dots <!--\; \dots \; -->,u_n{]}^{\mathrm{T}}$. I am trying to find a coordinate transformation matrix, $Q\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$, which is nonsingular, satisfying:
$$\begin{array}{r}\left[\begin{array}{c}0\\ ⋮<!--\; ⋮\; -->\\ 0\\ ||u||\end{array}\right]=Qu.\end{array}$$

Findthe Matrix T of the following linear transformation
$T:R2\left(x\right)-<!--\; -\; -->>R2\left(x\right)definedbyT(a{x}^2+bx+c)=2ax+b$

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal matrix represents a rotation (and possibly a reflection). Is it something similar about triangular matrices? Do they represent any specific type of transformation?

Consider ${\mathbb{K}}^n$, ${\mathbb{K}}^m$, both with the $||.|{|}_1$-norm, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.
Let $||T||=inf\{M\ge 0:||T(x)||\le M||x||\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->{\mathbb{K}}^n\}$ be the operator norm of a linear transformation $T:{\mathbb{K}}^n\to <!--\; \to \; -->{\mathbb{K}}^m$.
Show that the operator norm of the linear transformation $T$ is also given by:
$$||T||=max\{\underset{i=1}{\overset{m}{\sum <!--\; \sum \; -->}}|a_{ij}|,1\le j\le n\}=:||A|{|}_1$$
where $A$ is the transformation matrix of $T$ and $a_{ij}$ it's entry in the $i$-th row and $j$-the column.

Convert the following matrix:
$$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$
To the following:
$$\left[\begin{array}{cccccccc}0& 0& 1& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 0& 1& 0& 1\end{array}\right]$$

Real symmetric matrices $S_{ij}$ can always be put in a standard diagonal form $s_i{\mathrm{\delta <!--\; \delta \; -->}}_{ij}$ under an orthogonal transformation. Similarly, real antisymmetric matrices $A_{ij}$ can always be put in a standard band diagonal form with diagonal matrix entries $a_i\left(\begin{array}{cc}0& 1\\ -<!--\; -\; -->1& 0\end{array}\right)$ (with a $0$ diagonal entry when the dimension of the matrix is odd), again under an orthogonal transformation.

How to find a matrix of linear transformation $f:R^n\to <!--\; \to \; -->Ma{t}^{(n,n)}$.
Let's say we do $f:R^2\to <!--\; \to \; -->Ma{t}^{(2,2)}$ given by $f(x,y)=\left[\begin{array}{cc}x& 2y\\ x+y& x\end{array}\right]$
We calculate image of canonical basis $f(1,0)=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $f(0,1)=\left[\begin{array}{cc}0& 2\\ 1& 0\end{array}\right]$
Now the problematic part, ever since when caltulating matrix from vectors $R^n\to <!--\; \to \; -->R^m$ the approach is to transpose images of standard base $(f(1,0,...,0{)}^T|f(0,1,...,0{)}^T|f(0,0,...,1{)}^T)$. We can solve the $R\to <!--\; \to \; -->Mat$ problem by using $(A^T|B^T|C^T...)$, of course, but is there any way how to shrink the vector so we can succeed something like $(A^T|B^T|C^T...)$?

Let $T:V\to <!--\; \to \; -->W$ be a linear transformation of two finite dimensional vector spaces $V$,$W$ (both over a field $F$).
My assignment is to show that there exists a basis $B$ in $V$ and a basis $C$ in $W$ such that the transformation $F$ with respect to the bases $B$ and $C$ actually does have $A$ as a transformation matrix.
Since $V$,$W$ are vector spaces of course they have bases and I can do a transformation with respect to the bases from $V$ to $W$. But how do I show that (some specific?) bases exist such that the transformation with respect to these bases have a given a $A$ as a transformation matrix?
Assume $A$ is transformation matrix of $T$. Then we take one vector in $V$ expressed in (some basis) $B$ and when we transform it to some vector in $W$, expressed in (some basis) 𝐶. So $A$ does at least change the basis of a vector, but I since $V$ and $W$ might be of different dimension I don't really know where to go from here. Am I one the wrong track?

Linear transformation, T, such that:
T:$M_{22}$
$\left(\left[\begin{array}{cc}{x}_{11}& x_{12}\\ x_{21}& x_{22}\end{array}\right]\right)=\left[\begin{array}{cc}{x}_{12}-<!--\; -\; -->5x_{21}-<!--\; -\; -->x_{22}& -<!--\; -\; -->x_{11}-<!--\; -\; -->2x_{12}+3x_{21}+4x_{22}\\ -<!--\; -\; -->3x_{21}& -<!--\; -\; -->x_{11}-<!--\; -\; -->x_{12}+x_{21}+3x_{22}\end{array}\right]$
What is the matrix that represents this $M_{22}$->$M_{22}$ transformation?
Is it:
$\left[\begin{array}{cccc}0& 1& -<!--\; -\; -->5& -<!--\; -\; -->1\\ -<!--\; -\; -->1& -<!--\; -\; -->2& 3& 4\\ 0& 0& -<!--\; -\; -->3& 0\\ -<!--\; -\; -->1& -<!--\; -\; -->1& 1& 3\end{array}\right]$
If so, how could this be multiplied by a 2x2 matrix to give another 2x2 matrix. (2x2 matrices cannot multiply with 4x4 matrices).

Find the inverse, if it exists, for the given matrix
$\left[\begin{array}{cc}3& 2\\ 2& 5\end{array}\right]$

An augmented matrix of a system of equations has been transformed by row operations into the 3 equations in 3 variables matrix below. State the solution to this system as (x,y,z) Enter your answer as an ordered triple within parenthesis separated by commas (x,y,z). Avoid entering spaces and decimal points Use fractions if relevant.
$$\left[\begin{array}{ccccc}1& 0& 0& ⋮& -3\\ 0& 1& 0& ⋮& \frac{2}{3}\\ 0& 0& 1& ⋮& \frac{1}{4}\end{array}\right]$$

Let $A=\left[\begin{array}{cc}3& 2\\ 5& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-<!--\; -\; -->7& 9\\ 1& -<!--\; -\; -->9\end{array}\right]$
Find 3A-2B.

Perform the indicated matrix operation.
$\left[\begin{array}{ccc}2x+y& x-<!--\; -\; -->2y& 4x\\ 3x& 3y& x+y\end{array}\right]+\left[\begin{array}{ccc}5x& 8y& 3x+y\\ 5x+5y& x& 2x\end{array}\right]$

Find the inverse of the following matrix A, if possible. Check that $A{A}^{-<!--\; -\; -->1}=I$ and $A^{-<!--\; -\; -->1}A=I$
$A=\left[\begin{array}{cc}7& 4\\ 7& 7\end{array}\right]$
The inverse, $A^{-<!--\; -\; -->1}$ is ?

Find the inverse of the following matrix A, if possible. Check that $A\ast <!--\; \ast \; -->A^{-<!--\; -\; -->1}$ and $A^{-<!--\; -\; -->1}\ast <!--\; \ast \; -->A=I$
$A=\left[\begin{array}{cc}4& 8\\ -<!--\; -\; -->5& -<!--\; -\; -->10\end{array}\right]$
The inverse, $A^{-<!--\; -\; -->1}$, is A=?