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Recent questions in Matrix transformations

Hailie Blevins 2022-06-14

I have a matrices $A_{K \times M}$ and $B_{K \times N}$ I am looking for an orthogonal matrix $T_{M \times N}$ such that minimize:
${a r g m i n}_{T} | | A T - B | |$
(on other words: the best orthogonal transformation from A to B).
how can I find this T?

Sattelhofsk 2022-06-14

Given a matrix $A = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}]$ and bases to a the vector space $V$ :
$B = (v_{1}, v_{2}, v_{3}), B_{1} = (v_{1}, v_{1} + v_{2}, v_{1} + v_{2} + v_{3})$
Is A the transformation matrix between the basis $B$ to $B_{1}$ ?

Oakey1w 2022-06-14

Transformation Matrix of a dot product transformation
Let v be a arbitrary vector in R3 and T(x)=v.x. What is the matrix of the transformation T in terms of the components of v? It seems like trying to figure out the matrix using the equation T(x)=Ax does not work, as the left side is a scalar, and the other side is a matrix. Any idea

Brunton39 2022-06-14

Let's suppose $A, B$ are $n -$ order matrix, satisfying $A B = B A = O$ , $r (A) = r (A^{2})$ , prove that
$r (A + B) = r (A) + r (B)$

Celia Lucas 2022-06-14

If $T : R^{3} \to P^{2}$ where $T (a, b) = (a - b) t^{2} + (- a + b)$ , is $T$ a matrix transformation?

Yesenia Sherman 2022-06-13

A linear transformation between polynomial spaces $f : P_{2} (R) \to P_{2} (R)$ is given by
$f (p (x)) = 3 \cdot p (1) - x^{2} \cdot p (0) + (x - 1) \cdot p^{'} (1)$
Determine the transformation matrix with respect to the monomial basis
$(1, x, x^{2})$

Jeffery Clements 2022-06-13

Given $c$ in $R$ , define $T_{c} : R^{n} \to R$ by $T_{c} (x) = c x$ for all $x$ in $R^{n}$ . Show that $T_{c}$ is a linear transformation and find its matrix.
I don't understand the question. For $T_{c}$ is $c$ the matrix and we are supposed to find $c$ ? Would the matrix just be
$\begin{matrix} c & 0 & 0 . . .0 \end{matrix}$

mravinjakag 2022-06-12

The linear transformation $A : R^{2} \to R^{2}$ is given by the images of basis vectors: $A ((1, 1)) = (2, 1)$ and $A ((1, 0)) = (0, 3)$ .1. Find a matrix of linear transformation $A$ in the basis $(1, 1), (1, 0)$ .2. Find $A ((3, 2))$ 3. Find vector $x = (x_{1}, x_{2})$ such that the matrix $(\begin{matrix} - 6 & - 6 \\ 3 & 4 \end{matrix})$ is matrix of the linear transformation $A$ in the basis $(0, 3)$ ю

mravinjakag 2022-06-12

Finding the transformation matrix given the transformation
Given the Transformation $P (x_{1}, x_{2}, x_{3}) = (x_{2}, x_{3})$ , and I'm supposed to find the transformation matrix $A$ so that $A (x_{1}, x_{2}, x_{3}) = (x_{2}, x_{3})$
How do I do this? I managed to find it for a previous question through trial and error but it took me a while and I need a good mathematical way to do it

Tristian Velazquez 2022-06-10

Here $B$ , $V$ and $U$ are matrices:
Do relation $B (V + U) B^{- 1} = B V B^{- 1} + B U B^{- 1}$ hold true?

hawatajwizp 2022-06-10

A linear transformation $T : P_{2} \to P_{2}$ has matrix with respect to $S$ given by:
$[T] (S) = [\begin{matrix} 1 / 2 & - 3 & 1 / 2 \\ - 1 & 4 & - 1 \\ 1 / 2 & 2 & 1 / 2 \end{matrix}]$
How do you find $T (a + b x + c x^{2})$ ?

Jasmin Pineda 2022-06-08

Let $V$ be the vector space of polynomials of degree up to 2.
and $T : V \to V$ be a linear transformation defined by the type:
$T (p (x)) = p (2 x + 1)$
Find the matrix form of this linear transformation. The base to find the matrix is $B = {1, x, x^{2}}$

pokoljitef2 2022-06-07

Unimodular Matrix is a matrix, which has integer entries and determinant of $\in {+ 1, - 1}$ . That means the matrix is invertible over the integers $Z$ .
Given a unimodular matrix A for example:
$[\begin{matrix} 2 & - 2 & - 5 & - 4 & - 2 & - 3 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 3 & 1 & - 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]$
Need to convert to it to a matrix like this:
$[\begin{matrix} * & * & * & * & * & * \\ * & * & 0 & 0 & 0 & 0 \\ * & 0 & * & 0 & 0 & 0 \\ * & 0 & 0 & * & 0 & 0 \\ * & 0 & 0 & 0 & * & 0 \\ * & 0 & 0 & 0 & 0 & * \end{matrix}]$
where * denotes possible nonzero entries.

George Bray 2022-06-06

Let $K = R$ and $V := R [t]_{2}$ and let $F : R [t]_{2} \to R [t]_{2}, f \mapsto f (0) + f (1) \cdot (t + t^{2})$
Want to calculate $M_{S}^{S} (F)$ , the transformation Matrix of F with Basis S, being the standardbasis for the polynomialring $S = (1, t, t^{2})$
So we have to use the coordinate isomorphism:
$M_{S}^{S} (F) = (\begin{matrix} I_{S} (F (1)) & I_{S} (F (t)) & I_{S} (F (t^{2})) \end{matrix})$
Already know the solution, however I have no idea why the following matrix is the transformation Matrix of F.
$M_{S}^{S} (F) = (\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix})$
Looking at the transformation matrix we know that $F (1) = 1 + t + t^{2}$ ?

doodverft05 2022-06-06

I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the following transformation given:
$R_{2} [x] \to R_{2} [x]$
$(ϕ (p)) (x) = (x p (x + 1))^{'} - 2 p (x)$
We first have to find its matrix in basis
${1, x, x^{2}}$
which I know how to get. The transformation matrix result is:
$[\begin{matrix} - 1 & 1 & 1 \\ 0 & 0 & 4 \\ 0 & 0 & 1 \end{matrix}]$
How do I get the KERNEL and the IMAGE from it ?

kokoszzm 2022-06-06

Let there be a linear transformation going from $R^{3}$ to $R^{2}$ , defined by $T (x, y, z) = (x + y, 2 z - x)$ . Find the transformation matrix if base 1:
$⟨ (1, 0, - 1), (0, 1, 1), (1, 0, 0) ⟩$ ,
base 2: $⟨ (0, 1), (1, 1) ⟩$
An attempt at a solution included calculating the transformation on each of the bases in $R^{3}$ , (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of $B_{1}$ in $B_{2}$
Another point: if the basis for $R^{3}$ and $R^{2}$ are the standard basis for these spaces, the attempt at a solution is a correct answer.

polivijuye 2022-06-06

What is the transformation matrix?
In $R^{2}$ a basis is given $a = (a_{1}, a_{2})$ where:
$a_{1} = (1, - 1)$
$a_{2} = (0, 1)$
For $f : R^{2} \to R^{2}$ it is known:
$f (a_{1}) = - 6 \cdot a_{1}$
$f (a_{2}) = 1 \cdot a_{2}$
Determine the matrix of transformation with regards to the standard e-basis.

glycleWogry 2022-06-05

Similarity transformation of an orthogonal matrix
A transformation $T$ represented by an orthogonal matrix $A$ , so $A^{T} A = I$ . This transformation leaves norm unchanged.
I do a basis change using a matrix $B$ which isn't orthogonal , then the form of the transformation changes to $B^{- 1} A B$ in the new basis( A similarity transformation).
Therefore $B^{- 1} A B$ . $[B^{- 1} A B]^{T} = I$ .
This suggests that $B^{T} B = I$ which means it is orthogonal, but that is a contradiction.

aashi malhotra2022-06-04

if m² – 3m – 1 = 0, find the value of m²+1/m²

mitos3lmt53e 2022-06-04

a) Give the matrix for the Transformation $T : R^{2} \to R^{2}$ that first reflects points through the $x$ - axis and then reflects through the line $y = x$
b) How else can you describe this transformation?

If you are dealing with linear algebra, the chances are high that you will encounter various questions related to matrix transformation. Turning to matrix transformation examples, you will also encounter various geometric transformations, yet these will always be based on algebraic analysis and calculations. The answers that we have presented to various challenges will help you to compare our solutions with your unique matrix transformation example that deals with linear transformation and mapping. Visual assistance is also included and will be essential to see how these are built with the help of the column vectors.

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