MATH203-20S1 Linear Algebra
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Final Exam
MATH203-20S1
Linear Algebra
┌4(2) 0一
┐
1. Consider the matrix A = 
'''.
(a) State the Gerschgorin disks for A and sketch them in the complex plane. (b) State the eigenvalues of A.
(c) Starting with x| = [1, 1, 1, 1]T , perform 2 iterations of the Power Method on A.
(d) Which eigenvalue of A will the Power Method converge to and why? (e) State the rate of convergence of the Power Method for this A.
(f) Suppose that M and P are diagonalisable matrices with common eigenvectors (i.e., M = VDM V-| and P = VDP V-| ). Show that MP is diagonalisable.
(g) The matrix A above is diagonalisable, A = VDV-| (you do not need to show this). Suppose that B has the same eigenvectors as A, and let the eigenvalues of AB be λ| (AB) = λ2 (AB) = λ3 (AB) = λ4 (AB) = 1.
(i) State the eigenvalues of B .
(ii) What is the relationship between A and B?
2. Let A = ┌0(1) 1(2)
'2 一1
1(3)┐
1'.
(a) Find all solutions to Ax = ┌ ┐0(1)
'2'.
(b) What are the dimensions of the four fundamental subspaces of A?
(c) Is A invertible?
(d) Is A orthogonally diagonalisable?
(e) Find all solutions to Ax = ┌ 3一5┐
1
(f) Find a least squares solution to Ax = ┌ 3一5┐
' 1 '.
3. Consider the vectors
┌ ┐1(1) ┌ ┐2(0) ┌ 一11┐
w| = '(')0'(') , w2 = '(')1'(') , w3 = '(') 1 '(') ,
'0' '0' ' 0 '
and the subspace W = span|w| , w2 , w3 扌 ↓ R4 .
(a) Find an orthogonal basis for W .
(b) Find an orthogonal basis for WB .
(c) Compute the orthogonal projection of e| onto W .
(d) Find the distance between e| and W . (e) Find the distance between e| and WB .
(f) Compute the matrix of the linear map proji : R4 | R4 with respect to the basis (b| , . . . , b4 ), where the bi are the basis vectors computed in (3a) and (3b).
(g) Compute the matrix of the linear map proji I : R4 | R4 with respect to the basis (b| , . . . , b4 ), where the bi are the basis vectors computed in (3a) and (3b).
4. Determine whether the following statements are true or false. You must justify your answer to receive marks.
(a) If u, v ↓ Rn are linearly independent, then they are orthogonal.
(b) The eigenspaces of the matrix ┌ 1(1) 4(1)┐ are pairwise orthogonal.
(c) If A is an m × n matrix of rank m, then the equation Ax = b has a solution.
(d) The functions f (x) = x and g(x) = x 一 2/3 are orthogonal when considered as vectors in the inner product space C[0, 1] of continuous functions on [0, 1].
(e) The set of continuous functions f : [a, b] | R such that f (a) = 1 is a subspace of C[a, b].
(f) ┌3(3) 0(0)┐ is the best rank 1 approximation of ┌3(3) 1一1┐ .
2022-06-14