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

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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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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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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
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
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.

For Question 1.

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

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
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