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MATH2001/7000 ASSIGNMENT
创建日期：2024-06-01 16:15:17
所属分类：数学mathematics
标签： MATH2001
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MATH2001/7000 ASSIGNMENT

Due at 5:00pm 8 September. Marks for each question are shown. Total marks: 25
Submit your assignment online via the assignment 2 Gradescope submission link in Black-
board.
(1) (6 marks.) Let C[0, 1] have the inner product defined by
〈f ,g〉 =
∫ 1
0
f(x)g(x)dx, ∀f ,g ∈ C[0, 1].
Let U = span{x− 1, x+ 1, x+ x2} be a subspace of C[0, 1]. Find the best (i.e.
least squares) approximation of cosh(x) in the subspace U .
(2) (5 marks.) Let
A =
 4 2 20 4 0
0 −2 2
 .
(a) Find the eigenvalues and eigenvectors of A.
(b) Compute An, where n ≥ 1 is an integer.
(3) (5 marks.) Let A be a 3 × 3 real symmetric matrix and have eigenvalues λ1 =
2, λ2 = λ3 = 1. The eigenvectors corresponding to λ2 = λ3 = 1 are respectively
given by
v2 =
 10
1
 , v3 =
 12
−1
 .
Find the eigenvector of A corresponding to λ1 = 2 and the matrix A.
(4) (6 marks.) Let
A =
 3 2 42 0 2
4 2 3
 .
You are given that the characteristic equation of A is (λ+ 1)2(λ− 8) = 0, i.e. the
eigenvalues of A are λ = −1, 8.
(a) Find the eigenvectors of A.
(b) Are the eigenvectors of A corresponding to the eigenvalue λ = −1 orthogonal
with respect to the usual dot product? If not, apply the Gram-Schmidt
process to turn them into an orthonormal set of eigenvectors.
(c) Find an orthogonal matrix P that orthogonally diagonalizes A.
1
(5) (3 marks.) Let f(x, y, z) = x2 + y2 + z2 + 4b yz be a quadratic form, where b ∈ R.
Suppose that after the variable change, xy
z
 = P
 uv
w
,
where P is an orthogonal matrix, the quadratic form becomes f = u2 + 2v2 in
terms of the new variables u, v and w. Determine all possible values for b.
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