# Transformation matrix Definition and 8 Discussions

In linear algebra, linear transformations can be represented by matrices. If

T

{\displaystyle T}
is a linear transformation mapping

R

n

{\displaystyle \mathbb {R} ^{n}}
to

R

m

{\displaystyle \mathbb {R} ^{m}}
and

x

{\displaystyle \mathbf {x} }
is a column vector with

n

{\displaystyle n}
entries, then

T
(

x

)
=
A

x

{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some

m
×
n

{\displaystyle m\times n}
matrix

A

{\displaystyle A}
, called the transformation matrix of

T

{\displaystyle T}
. Note that

A

{\displaystyle A}
has

m

{\displaystyle m}
rows and

n

{\displaystyle n}
columns, whereas the transformation

T

{\displaystyle T}
is from

R

n

{\displaystyle \mathbb {R} ^{n}}
to

R

m

{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.

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1. ### I Dimension of a Linear Transformation Matrix

hi guys I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as ##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## , where ##\mu = (1 -1;-2 2)## and i found the matrix that corresponds to this linear...
2. ### I Transformation matrix for an expanding space

Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this: \begin{pmatrix}...
3. ### I Transformation of Tensor Components

In the transformation of tensor components when changing the co-ordinate system, can someone explain the following: Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...
4. ### Linear Algebra Follow up

Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...
5. ### Linear Algebra matrix linear transformation

Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...
6. ### How to form the transformation matrix for this

We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis. The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or...
7. ### QR Decomposition w/ Householder and Givens Transformations

Could anybody link me to some good examples on how to go about doing them? I honestly have no idea how to go about doing these two types of problems.
8. ### Function scales eigenvalues, but what happens to eigenvectors?

Statement: I can prove that if I apply a function to my matrix (lets call it) "A"...whatever that function does on A, it will do the same thing to the eigenvalues (I can prove this with a similarity transformation I think), so long as the function is basically a linear combination of the powers...