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Matrix Representation of a Linear Operator on N-dimensional Vector Space

1) Let A be a linear operator on n-dimensional vector space V if {see attachment}. For some {see attachment}, prove that matrix representation {see attachment} of A with respect to a basis ...

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1) Let A be a linear operator on n-dimensional vector space V if A 0 and A = 0
For some V, prove that matrix representation of A with respect to a basis
X=(x ,x ,................x ) is similar to the .
I have the answer but I do not understand very well
There is the answer
1) We have to prove that , A , A ,................ A .Is linearly independent which is easy.
2)The book said that the eigenvalues of A are all zero. This I don't know why.
3) The book said that the matrix of A under the basis is the matrix with all (i, i+1)-entries 1 and 0 elsewhere. This I don't understand.

2)Prove that if and is continuous in and
in this one we are studying inner product.

2) Prove that if A =A than A is similar to a diagonal matrix
I have the answer for this one too but I have some question
There is the answer
1. the only possible eigenvalues are 1 and 0. this I get it
2. Theonly eigenvalues of A are 0 and 1 Hence A is similar to a diagonal matrix of the form
D = (1,1,1,1,.....,0,0,.......0) where the number of 1's is equal to the rank of A. This I don't know why.

I will answer your these questions below.

For Question 1.

1) We have to prove that , A , A ,................ A is linearly independent.
In order to show that , A , A ...

Solution Summary

The matrix representation of a linear operator on n-dimensional vector space is investigated. Linear independence and eigenvalues are discussed. The solution is detailed and well presented.

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