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Recent questions in Matrix transformations

Tristan Meyers 2022-04-12

Consider a triangle that has vertices at A(1,2), B(2,3) and C(4,2). Reflect this triangle in the line through the origin which is inclined at $30^{\circ}$ to the positive x-axis. Find the vertices of the transformed triangle.
To solve this question, I rotate this triangle through $- 30^{\circ}$ first and then reflect it about x-axis and then rotate through $30^{\circ}$ . I am not sure if this way is correct.

Sheryl Awuor2022-03-28

Find k $k$ such that the following matrix M $M$ is singular.
M=⎡⎣⎢−225+k−5933−4114⎤⎦⎥

1183835381 2022-03-27

Let AB = Im, where A is an m×n matrix and B is an n×m matrix. If y ∼ MVN(μ,In) and BA is symmetric, find the distribution of yT BAy.

Find a 3x3 matrix A such that Ax⃗ =9x⃗ for all x⃗ in R

Determine which of the following transformations are linear transformations.

A. The transformation T defined by T(x1,x2,x3)=(1,x2,x3)T(x1,x2,x3)=(1,x2,x3)
B. The transformation T defined by T(x1,x2)=(2x1−3x2,x1+4,5x2)T(x1,x2)=(2x1−3x2,x1+4,5x2).
C. The transformation T defined by T(x1,x2)=(4x1−2x2,3|x2|)T(x1,x2)=(4x1−2x2,3|x2|).
D. The transformation T defined by T(x1,x2,x3)=(x1,x2,−x3)T(x1,x2,x3)=(x1,x2,−x3)
E. The transformation T defined by T(x1,x2,x3)=(x1,0,x3)

Kristina Mcclain 2022-02-02

Given Matrix transformation matrix
$A = (\begin{matrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{matrix})$
Are there vectors in $R^{3}$ that do not change under this transformation, i.e. such that $T (u) = u$ ? Explain why yes or why not.

bistandsq 2022-02-02

Lets say we have the following matrix
$M = [\begin{array}{cc} 4 & 3 \\ 4 & 3 \end{array}]$
Then I want to transform all numbers in the matrix with the following function:
$f (x) = {(x - 4)}^{2} + 2 x - 4$
What would be the correct notation for this?

jelentetvq 2022-02-02

From my working, P would equate to
$P = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]$
Now to find the transformation shown by $P^{3}$ , would I have to cube the matrix or is there something that I'm missing here?

Danitat6 2022-02-01

Consider the rectangle formed by the points (2,7),(2,6),(4,7) and (4,6). Is it still a rectangle after transformation by?
$A = (\begin{matrix} 3 & 1 \\ 2 & \frac{1}{2} \end{matrix})$
By what factor has its area changed ?

Jaylene Franco 2022-02-01

Determine transformation matrix
$f : P_{2} (R) \to P_{2} (R)$ is given by:
$f (p (x)) = 3 \cdot p (1) - x^{2} \cdot p (0) + (x - 1) \cdot p^{'} (1)$

Emerson Barnes 2022-02-01

Find the matrix representation of the linear transformation T that maps the input vector ${\vec{x}}_{1} = {(1, 1)}^{T}$ to the output vector ${\vec{y}}_{1} = {(2, - 3)}^{T}$ and maps the input vector ${\vec{x}}_{2} = {(1, 2)}^{T}$ to the output vector ${\vec{y}}_{2} = {(5, 1)}^{T}$

maliaseth0 2022-02-01

Writing a composite transformation as a matrix multiplication
I have two matrices, P and Q as follows:
$P = (\begin{matrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & - \frac{1}{2} \end{matrix})$
$Q = (\begin{matrix} - \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{matrix})$

Addison Gross 2022-01-31

How to find the linear transformation associated with a given matrix?
It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation $T : V \to W$ can be represented as a matrix, but since this is an isomorphism between $L (V, W)$ and $M_{m \times n}$ where the latter represents the space $m \times n$ matrices on the same field in which are defined respectively vector spaces.

Berasiniz1 2022-01-31

Show that the given transformation is linear, by showing it is a matrix transformation:
$F [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} x = y \end{matrix}]$

bistandsq 2022-01-31

Let $T : R^{3} \to R^{3}$ be the linear transformation such that
$T [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} - 2 \\ 5 \\ - 2 \end{matrix}], T [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} 4 \\ 1 \\ 4 \end{matrix}], T [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}]$
a) Find a matrix A such that $T (x) = A x$ for every $x \in R^{3}$
b) Find a linearly independent set of vectors in $R^{3}$ that spans the range of T

Seamus Kent 2022-01-31

1) Calculate the transformation matrix
2) Calculate the dimension of the kernel of the transformation, justify.
$T : R^{3} \to R^{3}$ is a linear transformation such that:
$T ({\vec{e}}_{1}) = {\vec{e}}_{1} - {\vec{e}}_{2} + 2 {\vec{e}}_{3}$
$T ({\vec{e}}_{1} + {\vec{e}}_{2}) = 2 {\vec{e}}_{3}$
$T ({\vec{e}}_{1} + {\vec{e}}_{2} + {\vec{e}}_{3}) = - {\vec{e}}_{2} + {\vec{e}}_{3}$

Ella Bradshaw 2022-01-31

Can a triangle be represented as a matrix?
Triangle1 (1, 1), (1, 2), (3, 1)
Triangle2 (1, 3), (1, 6), (3, 3)
Find the matrix which represents the stretch that maps triangle T1 onto triangle T2.

veceriraby 2022-01-31

a) Find the standard matrix of the linear operator $T : R^{2} \to R_{2}$ given by the orthogonal projection onto the vector (1,-2).
b) Given the linear transformation $T : R^{2} \to R_{3}$ such that
Find T(2,3).

Celia Horne 2022-01-31

Find a reflection matrix about a given line, using matrix multiplication and the idea of composition of transformations.
The line of: $y = - \frac{2 x}{3}$ , all in $R^{2}$

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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