연세대 선형대수학 족보 2학기-선대시험-3차기말-모범답안

Problem 1. Indicate whether the statement is true(T) or (5) If A is a symmetric matrix, then eigenvectors from dierent eigenspaces are orthogonal. (T) false(F). Justify your answer. [each 3pt] (1) If T : Rn → Rn is a linear operator, and if [T]B = [T]B with respect to two bases B and B for Rn , then B = B . (F) Solve If T is a zero operator, then [T]B = O for any basis for R . So [T]B = [T]B but B = B . So (λ1 λ2)(x1 x2) = 0 and thus x1 x2 = 0.
n

solve Suppose that x1 ∈ Eλ1 and x2 ∈ Eλ2 are eigenvectors from dierent eigenspaces. Then, (λ1 x1) x2 = (Ax1) x2 = x1 (AT x2)

= x1 (Ax2) = x1 (λ2 x2)

(2) If V and W are distinct subspaces of Rn with the same dimension, then neither V nor W is a subspace of the other. (T) Solve With out loss of generality, if V is a subspace of W . Since dim(V) = dim(W) and V is a subspace of W , V = W . It is a contradiction. Problem 2. Indicate whether the statement is true(T) or false(F). [each 2pt] (1) If A = U ΣV T is a singular …