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無年度 - 主題課程_對角化：特徵值、特徵向量、特徵空間#107848
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105中興電機
(10pts) Suppose A and B are two matrices with size m✕n and n✕m resbectively. Show that the nonzero eigenvalues of AB and BA are the same.
其他申論題
(b) (6%) Please find the eigenvalues of 2A.
#462092
(c) (6%) Please find the eigenvalues of A2.
#462093
109中興電機 (15 pts) The eigenvalues of A and AT are the same, because det(A- λ l) -det(A- λ l)T= det(AT- λ l). By coming up with a 2 X 2 counter-example, show an example that the cigenvectors of A and ATneed not be the same.
#462094
106中興電機 (10pts) Suppose A is a real and symmetric matrix with order n, and has a repeated eigenvalue. Then for every i in {1,2, ... n}, there exists an eigenvector whose ith component is 0.
#462095
Let .Show that is linear combination of A and I2.
#462097
Suppose that T: R2→R3 is a linear transformation such that and Determine for any in R2.
#462098
Let T be a linear operator on R3 such that T, Please find the standard matrix of T.
#462099
(a) Find .
#462100
(b)Find
#462101
Let Y be the vector space of 2x2 mattices with real entries, and P3 the vector space of real polynomials of dogree 3 or less. Detine the linear transformation T: V → P3 by == 2a+(6-d)t-(a+c)x2+(a+6-c-d)x3. Find the rank and nullity of T.
#462102
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