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## Homework Statement

1. Let [tex]\theta[/tex] be a real number. Prove that the following two matrices are similar over the field of complex numbers:

[tex]\left [\stackrel{cos \theta}{sin \theta} \stackrel{-sin \theta}{cos \theta} \right][/tex] and [tex] \left [\stackrel{ e^{i \theta} }{0} \stackrel{0}{e^ {-i \theta}} \right][/tex]

2. Let W be the space of all nx1 column matrices over R. If A is an nxn matrix over R, then A defines a linear operator La on W through left multiplication : La (X) = AX. Prove that every linear operator on W is left multiplication by some matrix A.

Now, if T,S be operators such that T

^{n}= S

^{n}= 0 but T

^{n-1}[tex]\neq[/tex] 0, S

^{n-1}[tex]\neq[/tex] 0 . Prove that T and S both have the same matrix A for some basis B for T and B' for S.

Similarly show that if M and N are nxn matrices such that M

^{n}= N

^{n}= 0 but M

^{n-1}= N

^{n-1}[tex]\neq[/tex] 0, then M and N are similar.

## The Attempt at a Solution

Sum number 1: I'm not sure how to start this.

Sum number 2: The first part is okay. I can always find/make some matrix A such that the column space of A is the range of La.

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