Matrix Transformation Examples and Practice Problems

Find the transformation matrix R that relates the (orthonormal ) standard basis of ${\mathbb{C}}^3$ to the orthonormal basis obtained from the following vectors via the Gram Schmidt process:
a1> = $\left(\begin{array}{c}1\\ i\\ 0\end{array}\right)$
a2> = $\left(\begin{array}{c}0\\ 1\\ -<!--\; -\; -->i\end{array}\right)$
a3> = $\left(\begin{array}{c}i\\ 0\\ -<!--\; -\; -->1\end{array}\right)$

Finding inverse matrix $A^{-<!--\; -\; -->1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A.
$A=\left(\begin{array}{cccc}1& 3& 9& 27\\ 3& 3& 9& 27\\ 9& 9& 9& 27\\ 27& 27& 27& 27\end{array}\right)$
$B=\left(\begin{array}{cccc}1& 3& 9& 27\\ 1& 1& 3& 9\\ 1& 1& 1& 3\\ 1& 1& 1& 1\end{array}\right)$
I found that that $B^{-<!--\; -\; -->1}$ is
$B^{-<!--\; -\; -->1}=\left(\begin{array}{cccc}-<!--\; -\; -->\frac{1}{2}& \frac{3}{2}& 0& 0\\ \frac{1}{2}& -<!--\; -\; -->2& \frac{3}{2}& 0\\ 0& \frac{1}{2}& -<!--\; -\; -->2& \frac{3}{2}\\ 0& 0& \frac{1}{2}& -<!--\; -\; -->\frac{1}{2}\end{array}\right)$
I don't know how to continue. What rule do I use to find $A^{-<!--\; -\; -->1}$?

Prove that pre-multiplying a matrix $A_m$ by the elementary matrix obtained with any matrix elementary line transformation $I_m\underset{l_1\leftrightarrow <!--\; \leftrightarrow \; -->l_2}{⟶<!--\; ⟶\; -->}E$ is the same as applying said elementary line transformation on the matrix $A_m$

$T(p(x))={\int <!--\; \int \; -->}_0^{1}p(x)dx.$
(a) Show $T$ is a linear transformation.
(b) Compute $\mathcal{N}(T).$ Is $T$ one-to-one?
(c) Show that $T$ is onto.
(d) Let $B$ be the standard basis for ${\mathcal{P}}_2$ and let $B^{\mathrm{\prime <!--\; \text{'}\; -->}}=\{1\}$ be a basis for $\mathbb{R}$. Find $[T{]}_B^{B^{\mathrm{\prime <!--\; \text{'}\; -->}}}$.
(e) Use the matrix found in part (d) to compute $T(-<!--\; -\; -->x^2-<!--\; -\; -->3x+2)$

Find the matrix A for the linear transformation T relative to the bases ${\displaystyle B=\{1,x,x^2\}}$ and ${\displaystyle B\prime =\{1,x,x^2,x^3\}}$ such that ${\displaystyle T\left(\overrightarrow{x}\right)=A\overrightarrow{x}}$

Find the transformation matrix:
$F:{\mathbb{R}}_3[x]\mathbb{]}\to <!--\; \to \; -->{\mathbb{R}}_3[x]$
$F(v)=\frac{d^{2}v}{d{v}^2}$
Basis: $1,x,x^2,x^3$ and ${\mathbb{R}}_3[x]$ - the set of all third degree polynomials of variable $x$ over $\mathbb{R}$ Assume that all coefficients of the polynomials are $1$
The first thing that springs to my mind is to calculate this derivative by hand, and so we got
$\frac{d^{2}y}{d{y}^2}=2+6x$
Now, we need to put these values - $2$ and $6$ in such a matrix that - when multiplied by the basis vector -will give us $2+6x$ But there are many ways I can think of, for example
$\left[\begin{array}{cccc}0& 0& 2& 0\\ 0& 0& 0& 6\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
Or maybe
$\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 6& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
Because both of them, when multiplied by$\left[\begin{array}{c}1\\ x\\ x^2\\ x^3\end{array}\right]$
Will give the correct answer. Thus, what is the correct way to solve this?

Consider in ${\mathbb{R}}^2$ the set of points satisfying the equation (you can use matlab for this). How are these points transformed by the following matrix:
$\left[\begin{array}{cc}2& -<!--\; -\; -->1\\ -<!--\; -\; -->1& 1\end{array}\right]$
[Show the transformed set and plot it – you can use matlab again]
I could be mistaken but would the matrix be
$\left[\begin{array}{cc}2& -<!--\; -\; -->1-<!--\; -\; -->2\end{array}\right]?$
After that I am very stuck. Thanks

If the transformation is from ${\mathbb{R}}^3\to <!--\; \backslash rightarrow\; -->\mathbb{R}$ is
$T\{a,b,c\}={\int <!--\; \int \; -->}_0^{\mathrm{\pi <!--\; \pi \; -->}}2a{e}^t+2b\sin <!--\; \; -->(t)+3c\cos <!--\; \; -->(t)dt$
How to find the standard matrix?

In an affine transformation $x\mapsto <!--\; \mapsto \; -->Ax+b$, $b$ represents the translation; but what does the matrix A represent exactly? In a 2D example, $A$ is a $2\times <!--\; \times \; -->2$ matrix, but what does each term represent?

If I take a particular number, for example 21, and write down all possible matrices whose determinant is 21, will all these matrices represent the same transformation with different system of bases? Consider all endomorphisms.

Transformation matrix for matrix indices to cartesian coordinates
(1,1) (1,2) (1,3)(2,1) (2,2) (2,3)(3,1) (3,2) (3,3)
This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation would be:
(-1,1) (0,1) (1,1)(-1,0) (0,0) (1,0)(-1,1) (0,-1) (1,-1)
Say, I have any square matrix with dimension-N, where N is odd. I need a generic transformation matrix such that I can get a vector as cartesian coordinates from matrix indices. Does such a function already exist? How should I go ahead in implementing this?

Let $v\in <!--\; \in \; -->T_2(V)$ be a bilinear form over finite space V. Let T be a Linear transformation $V\to <!--\; \backslash rightarrow\; -->V$. We define: $v_T(x,y)=v(T(x),y)$
Assuming $v$ is nondegenerate, let us have another bilinear form $\mathrm{\xi <!--\; \xi \; -->}\in <!--\; \in \; -->T_2(V)$. Prove that there exists exactly one transformation $T$ so $\mathrm{\xi <!--\; \xi \; -->}=v_T$.

Let's say we define the orientation of the ellipsoid from its major axis (the largest axis of the ellipsoid). Assuming the 3 axes of the ellipsoid to be on the three coordinates with lengths of $a$, $b$ and $c$ along each axis (with $a\ge <!--\; \ge \; -->b$ and $a\ge <!--\; \ge \; -->c$), then only a single affine transformation (to a sphere) is necessary:
$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]$
Now, if we first rotate the major axis by $\mathrm{\theta <!--\; \theta \; -->}$ from the first axis towards the second axis, and then rotate it by $\mathrm{\varphi <!--\; \varphi \; -->}$ from the (rotated) first axis towards the third axis, the combined affine transformation becomes:
$\left[\begin{array}{cccc}a& 0& 0& 0\\ 0& b& 0& 0\\ 0& 0& c& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\mathrm{\theta <!--\; \theta \; -->}& -<!--\; -\; -->sin\mathrm{\theta <!--\; \theta \; -->}& 0& 0\\ sin\mathrm{\theta <!--\; \theta \; -->}& cos\mathrm{\theta <!--\; \theta \; -->}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}cos\mathrm{\varphi <!--\; \varphi \; -->}& 0& -<!--\; -\; -->sin\mathrm{\varphi <!--\; \varphi \; -->}& 0\\ 0& 1& 0& 0\\ sin\mathrm{\varphi <!--\; \varphi \; -->}& 0& cos\mathrm{\varphi <!--\; \varphi \; -->}& 0\\ 0& 0& 0& 1\end{array}\right]$
Is the multiplied matrix (from left to right) the correct affine transformation that must go into the equations in the links above?

Suppose $T$ is an element of $L(P_3(\mathbb{R}),P_2(\mathbb{R}))$ is the differentiation map defined by $Tp=p^{\prime }$. Find a basis of $P_3(\mathbb{R})$ and a basis of $P_2(\mathbb{R})$ such that the matrix of T with respect to these basis is
$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)$

I have a rectangle with center $(x_1,y_1)$ and sides $a,b$ where side $a$ is parallel to axis $Ox$. I want to find a transformation matrix that:
a) converts this rectangle into a square with the same center and side $d$.
b) reflects the rectangle with mirror axis the line $y=sx+c$.

A linear transformation $T:<!--\; :\; -->{\mathbb{R}}^3\to <!--\; \backslash rightarrow\; -->{\mathbb{R}}^2$ whose matrix is
$\left(\begin{array}{ccc}1& 3& 3\\ 2& 6& -<!--\; -\; -->3.5+k\end{array}\right)$
is onto if and only if $k\ne <!--\; \ne \; -->$

Let $V$ be inner product space.
Let $e_1,...,e_n$ an orthonormal basis for $V$
Let $z_1,...,z_n$ an orthonormal basis for $V$
I have to show that the matrix represents the transformation matrix between $e_1,...,e_n$ to $z_1,...,z_n$ is unitary.

If
$A=\left[\begin{array}{ccc}1& -<!--\; -\; -->1& 2\\ -<!--\; -\; -->2& 1& -<!--\; -\; -->1\\ 1& 2& 3\end{array}\right]$
is the matrix representation of a linear transformation
$T:P_2(x)\to <!--\; \backslash rightarrow\; -->P_2(x)$
with respect to the bases $\{1-<!--\; -\; -->x,x(1-<!--\; -\; -->x),x(1+x)\}$ and $\{1,1+x,1+x^2\}$ then find T. What is the procedure to solve it?

If I am required to compute the full transformation matrix compromising of the following sequence of operations:
rotation by 30 degrees about x-axis
translation by 1, -1, 4 in x, y and z, respectively
rotation by 45 degrees about y axis
Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?