Matrix of linear transformation

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Homework Statement

Find the matrix of the transformation:

[tex]T: R^{2} \rightarrow R^{2x2}[/tex]

[tex]
\[
T(a,b) =
\left[ {\begin{array}{cc}
a & 0 \\
0 & b \\
\end{array} } \right]
\][/tex]



Homework Equations





The Attempt at a Solution


I choose the standard bases for [tex]R^{2}[/tex] and [tex]R^{2x2}[/tex] and call them b and b' respectively.

[tex]T(1,0) = 1e_{1} + 0e_{2} + 0e_{3} + 0e_{4}[/tex]
[tex]T(0,1) = 0e_{1} + 0e_{2} + 0e_{3} + 1e_{4}[/tex]

This gives me a matrix of

[tex]
\[
\left[ {\begin{array}{cc}
1 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 1 \\
\end{array} } \right]
\][/tex]

However, this doesn't work when I multiply by the column vector (a,b). I get a column vector of (a, 0, 0, b) instead of a 2x2 matrix. What's going on?
 

Answers and Replies

  • #2
radou
Homework Helper
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If your R^2x2 is the subspace of all 4x4 matrices of this given form, what is its dimension? You have too many basis vectors here.
 
  • #3
70
0
If your R^2x2 is the subspace of all 4x4 matrices of this given form, what is its dimension? You have too many basis vectors here.
R^2x2 is the space of all 2x2 matrices. It has 4 basis vectors with 1 in the i,j position and zeroes everywhere else.
 

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