Dimension of a Linear Transformation Matrix

In summary, when finding the matrix of a linear transformation from a vector space to itself, the dimension of the vector space will determine the size of the matrix. In this case, since both the domain and the range are ##M_2(\mathbb{R})##, the matrix should be 4 x 4. This matrix can then be used to act on matrices in the ##M_2(\mathbb{R})## space, either by multiplying from the left or from the right, depending on the chosen interpretation.
  • #1
patric44
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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
 
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  • #2
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.
 
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  • #3
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.
 
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  • #4
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
 
  • #5
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*}
 
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What is the dimension of a linear transformation matrix?

The dimension of a linear transformation matrix refers to the number of rows and columns in the matrix. It represents the number of inputs and outputs of the transformation.

How is the dimension of a linear transformation matrix related to the number of variables in a system?

The number of variables in a system is equal to the dimension of the input vector, which is also equal to the number of columns in the transformation matrix.

What is the significance of the dimension of a linear transformation matrix?

The dimension of a linear transformation matrix determines the number of inputs and outputs of the transformation, and thus, the size of the vector space that the transformation maps between.

Can the dimension of a linear transformation matrix change?

No, the dimension of a linear transformation matrix is fixed and cannot change. It is determined by the number of rows and columns in the matrix, which cannot be altered without changing the transformation itself.

How can the dimension of a linear transformation matrix be used to determine if a system is solvable?

If the dimension of the transformation matrix is equal to the number of variables in the system, then the system is solvable. If the dimension is greater than the number of variables, the system is overdetermined and may not have a unique solution. If the dimension is less than the number of variables, the system is underdetermined and may have infinite solutions.

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