I Dimension of a Linear Transformation Matrix

patric44
Messages
308
Reaction score
40
TL;DR Summary
does the matrix of the linear transformation should have the same dimension of the vector space?
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 transformation this 4*4 matrix , given by
$$\phi =(1\;0\;-2\;0;0\;1\;0-2;-1\;0\;2\;0;0 -1\;0\;2) $$
is there is some wrong here or the matrix should be 4*4
 
Physics news on Phys.org
patric44 said:
Summary:: does the matrix of the linear transformation should have the same dimension of the vector space?

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 transformation this 4*4 matrix , given by
$$\phi =(1\;0\;-2\;0;0\;1\;0-2;-1\;0\;2\;0;0 -1\;0\;2) $$
is there is some wrong here or the matrix should be 4*4
The matrix should be 4 x 4, since your transformation is a map from ##M_2## to itself. ##M_2##, the space of 2 x 2 matrices, is of dimension 4, and any basis for this space will need to have 4 elements.
 
A linear transformation ##\Phi: V\longrightarrow W## of finite-dimensional vector spaces, say ##\dim V = n## and ##\dim W = m## has a matrix representation as an ##m \times n## matrix, ##n## columns and ##m## rows.

We have here ##V=W=\mathbb{R}^2,## the vector space of linear functions ##\varphi : \mathbb{R}^2 \longrightarrow \mathbb{R}^2##. Since ##\dim \mathbb{R}^2 = 2##, the dimension of ##\mathbb{M}_2(\mathbb{R})## is ##2\cdot 2= 4.##

In case ##V=W=\mathbb{M}_2(\mathbb{R})## we get ##\dim V = \dim W=\dim \mathbb{M}_2(\mathbb{R})=4,## we have ##\dim \mathbb{M}_4(\mathbb{R})=4\cdot 4=16## and ##\Phi## is represented by a ##4 \times 4## matrix.
 
thanks guys i think i got it now, but i still have a question : how iam going to act with a 4*4 matrix on matrices that belong to ##M_{2}(R)## space
 
If you use them as a linear transformation of ##\mathbb{R}^2## then you simply consider the ##2\times 2## matrix.

If you want to deal with all those linear transformations and deal with ##\mathbb{M}_2(\mathbb{R})## as vector space instead, then you have to choose some ordered basis to make those matrices a linear vector. E.g. if ##e_{ij}## is the matrix with ##1## in the ## i ##-th row and ##j##-th column, and zeros otherwise, then ##(e_{11},e_{12},e_{21},e_{22})## would be such a bases.

A ##2 \times 2## matrix would be
##\begin{bmatrix}
2&3\\4&5
\end{bmatrix}=2e_{11}+3e_{12}+4e_{21}+5e_{22}=(2,3,4,5)##

A linear transformation between those, e.g. ##M_{\Phi}=\begin{bmatrix}
2&3&0&1\\0&0&0&5\\0&0&1&4\\0&0&0&0
\end{bmatrix}##
would map
\begin{align*}
e_{11}&\longmapsto 2e_{11}+3e_{12}+1e_{22}\\
e_{12}&\longmapsto 5e_{22}\\
e_{21}&\longmapsto 1e_{21}+4e_{22}\\
e_{22}&\longmapsto 0
\end{align*}Edit: This interpretation corresponds to ##\vec{v} \longmapsto \vec{v}\cdot M_{\Phi}##. If we choose matrix multiplication from the left, i.e. ##\vec{v} \longmapsto M_{\Phi}\cdot \vec{v}## then
\begin{align*}
e_{11}&\longmapsto 2e_{11}\\
e_{12}&\longmapsto 3e_{11}\\
e_{21}&\longmapsto 1e_{21}\\
e_{22}&\longmapsto 1e_{11}+5e_{12}+4e_{21}
\end{align*}
 
Last edited:
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...

Similar threads

Back
Top