(c) (6%) Please find the eigenvalues of A². 

(a) (6%) Plcase find the eigenvalues of.

(b) (6%) Please find the eigenvalues of 2A.

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.

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.

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.

(12%) Let V be a finite-dimensional vector space and T :V →V be linear. Suppose rank(T) = rank(T2). Prove that R(T) ∩N(T) = {0}.

(13%) Given a vector space V over F. Define the dual space of V* of V as the set of all functions (also known as linear functionals) from V to F, i.e, V* {f|f : V →F}. It is obvious that V* is itself also a vector space with the addition + :V*'x V*→ V* and scalar multiplication * : F x V*→  V* defined as pointwise addition as well as pointwise scalar multiplication. Given any linear transformation T : V→  W. The transpose Tt is a linear transformation from W* to V* defined by Tt(f) = fT for any f W*. For every subset S of V, we define the annihilator Suppose V, W are both finite-dimensional vector spaces and T : V → W is linear. Prove that .N(Tt) = (R(T))0.

(10%) The nullities of the matrices BBT - λ/ for  λ= 0, 1,2,3, 4  are______,________,_________,_______ respectively. 

(d) (2%) Compute dim(N(Bt A)).

(c) (3%) Compute rank(AtAAAt).

無年度 - 主題課程_向量空間：秩(rank)、零度(nullity)、維度#108952

無年度 - 主題課程_向量空間：基底和維度#108872

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無年度 - 主題課程_向量空間：線性獨立和線性相依#108250

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無年度 - 主題課程_向量空間：向量空間和子空間#107951

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無年度 - 主題課程_理工學院機率論：單一隨機變數#107935

無年度 - 主題課程_行列式和線性方程式：線性方程式#107934

無年度 - 主題課程_線性映射：基底變換和座標變換#107933