Linear Transformation of Matrix

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  • #1
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Homework Statement



Let A[itex]_{2x2}[/itex] have all entries=1 and let T: M[itex]_{2x2}[/itex][itex]\rightarrow[/itex]M[itex]_{2x2}[/itex] be the linear transformation defined by T(B)=AB for all B[itex]\in[/itex]M[itex]_{2x2}[/itex]

Find the matrix C=[T]s,s, where S is the standard basis for M[itex]_{2x2}[/itex]

My solution:

Standard basis for M[itex]_{2x2}[/itex]={(1,0),(0,1)}
T(1,0)=(1,1)
T(0,1)=(1,1)
[T]s,s=(1,1;1,1)

I'm not sure how correct this is. Any advice would be appreciated.
 

Answers and Replies

  • #2
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Still confused on this one
 
  • #3
Dick
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Still confused on this one

The standard basis for M_2x2 is four matrices [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[1,0]] and [[0,0],[0,1]]. You can express any other matrix as a linear sum of those. Now take a look at the problem again.
 
  • #4
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Doesn't this still produce the same vector?
 
  • #5
Dick
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Doesn't this still produce the same vector?

What vector? What is A times the first basis matrix?
 
  • #6
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Sorry, the same matrix

A times first basis matrix is

[[1,0][0,0]]
then
[[0,0][1,0]], [[0,1][0,0]], and [[0,0][0,1]]
 
  • #7
Dick
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Sorry, the same matrix

A times first basis matrix is

[[1,0][0,0]]
then
[[0,0][1,0]], [[0,1][0,0]], and [[0,0][0,1]]

[[1,1],[1,1]]*[[1,0],[0,0]] isn't equal to [[1,0],[0,0]].
 
  • #8
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It appears it isn't

[[1,0][1,0]]

then

[[0,1][0,1]], [[1,0][1,0]], and [[0,1][0,1]]
 
  • #9
Dick
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It appears it isn't

[[1,0][1,0]]

then

[[0,1][0,1]], [[1,0][1,0]], and [[0,1][0,1]]

Ok, so work on what the matrix C should be. It should be 4x4 since you have four basis elements.
 
  • #10
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Would this be

[[1,0,0,1],[1,0,0,1],[1,0,0,1],[1,0,0,1]]?
 
  • #11
Dick
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Would this be

[[1,0,0,1],[1,0,0,1],[1,0,0,1],[1,0,0,1]]?

It would depend on which column represents which basis element. You should spell that out. But no I don't think that's it. How did you conclude that?
 

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