Final Exam, 2021 Spring Semester
Multiple choices. (More than one can be true. Penalty of 1/2 of the points for each
wrong answer.)
1. (5×2 = 10%) Which statements are true? (b, d)(b)
a. If P projects onto the column space of a matrix A, then I- P projects onto the
null space of A.
b. For a tall matrix 𝐴, the least-squares solution of the linear equation system
𝐴𝒙 = 𝒃 minimizes the norm ‖𝐴𝒙 − 𝒃‖.
c. Three orthogonal none-zero vectors can be constructed by the Gram-Schmidt
procedure from the vectors a1 = [1, -1, 0, 0]T, a2 = [0, 1, -1, 0]T, and a3 = [1, 0,
-1, 0]T.
d. If a 4 by 3 matrix 𝐴 has three orthogonal columns, then 𝐴𝑇 𝐴 is an identity
matrix.
cos 𝜃
e. In 𝐑2 , the rotation matrix [
sin 𝜃
− sin 𝜃
] can rotate the vector [4, 3]T to the
cos 𝜃
vector [-3, 4]T if 𝜃 = −𝜋/2.
2. (5×2 = 10%) Which statements are true? (a, b, d, e) (a, b, d)
0 ×
a. If 𝐴 = [× ×
0 ×
0
×], regardless of the values of the five non-zeros marked by
0
×’s, then det A = 0,
b. If A and B are both invertible and AB exists, then AB is invertible.
c. If a 3 by 3 matrix has three eigenvalues 0, 1, 2, then its rank is 3.
cos 𝜃
d. The rotation matrix [
sin 𝜃
− sin 𝜃
] in 𝐑2 does not have eigenvectors in 𝐑2 ,
cos 𝜃
unless 𝜃 is the multiple of 𝜋.
e. A matrix with real eigenvalues and orthogonal eigenvectors is symmetric.
3. (5×2 = 10%) Which statements are true? (a, d) (a)
0.03
0
a. The 3 by 3 matrix [ 0
0.02
0
0
0
0 ] is positive definite.
0.01
b. In the SVD A = UVT, the columns of U and V are mutually orthogonal.
c. If A is square but not invertible, then A cannot have SVD.
1 −1
d. The matrix [
] is similar to the identity matrix.
0 1
e. The rank-one matrix A = uvT yields an SVD with 1 as the singular value.
4. (10%) Find the determinant of a 2n×2n block-diagonal matrix A where the entries
A2k-1,2k-1 A2k-1,2k
1 1
following [
]=[
] (for k = 1, …, n) and zeros everywhere
A2k,2k-1
A2k,2k
0 1
else. The value is
(a) 0; (b) 1; (c) n; (d) n2; (e) None of the shown values.
(b)
5. (10%) The 2 by 2 matrix 𝐴 = [
[
1.5 -0.5
]. Find 𝐴𝑘 as 𝑘 → ∞. The answer is (a)
-0.5 1.5
∞
1 0
0 0
0.5 0.5
]; (b) [
]; (c) [
]; (d) [
∞
0 1
0 0
0.5 0.5
∞
]; (e) None of the shown matrices.
∞
(c) (e)
6.
1
(10%) Define the quadratic function y = 2xTATAx + d where A is an m by n
full-ranked matrix and d is a constant. Which statements are true?
(b, d, e)
a. The size of x and y must be the same.
b.
c.
d.
e.
If d ≥ 0, then y ≥ 0.
The function has a maximum but not a minimum.
The extremum (maximum or minimum) is d.
The matrix ATA is not strictly positive definite.
7. (5×2 = 10%) Which statements are true? (b, c, e) (b, c)
a. If S and T are linear transformations, then the combination S(T(v)) is a
quadratic transformation.
b. Every linear transformation from 𝐑2 to 𝐑2 takes squares to parallelograms.
c. The kernel of the rotation in 𝐑2 is the zero vector.
d. The range of a projection in 𝐑2 cannot be a plane.
e. In 𝐑2 , the translation T(v)= v+b with b being a constant vector is linear.
8. (10%) Let Pn = {p(x) = c0 + c1x + … + cnxn} be the set of all n-th order
polynomials with real-valued coefficient ci. Note that Pn can be regarded as a
vector space. Which statements are true?
(a, c)
a. The set B = {1, x, x2, x3} can be used as a basis for P3.
b. The operation O(p(x)) = xp(x) can be viewed as a linear transformation from
Pn+1 to Pn.
c. Define the inner product in Pn as ⟨𝑝(𝑥), 𝑞(𝑥)⟩ = ∑𝑛𝑖=0 𝑐𝑖 𝑑𝑖 where 𝑐𝑖 is the
coefficient for 𝑝(𝑥) and 𝑑𝑖 for 𝑞(𝑥). Then 100 + 10x + x3 and 1 -10x + x2 are
orthogonal to each other.
d. The matrix representation of the operation O(p(x)) = xp(x) is a diagonal matrix
if the basis {1, x,… , xn+1} is used.
e. Take derivative of a polynomial cannot be viewed as a linear transformation
because it changes the order of the polynomial.
9. (10%) For a linear transformation 𝑇: 𝐑𝑛 → 𝐑𝑚 . Which statements are true?
(a, b, c, e) or (a, c, e)
a. If the range of 𝑇 is 𝐑𝑚 , then the standard matrix of 𝑇 has full row rank.
b. If 𝑛 = 𝑚 and the standard matrix of 𝑇 has n linear eigenvectors, then it can
be diagonalized. 同學問題目沒有說 eigenvector 有限性獨立 false
c. Let B={[1]} and B’={[1 1]T, [0 1]T} be the basis for 𝐑1 and 𝐑2 respectively,
then the matrix representation of 𝑇([𝑥]) = [
𝑥
1
], according to B and B’, is [ ].
2𝑥
1
d. Not every linear transformation 𝑇: 𝐑𝑛 → 𝐑𝑚 has a matrix representation.
e. Let the matrix representation 𝑇 of is A, then the matrix representation of the
function 𝑆(𝒙) = 𝑇(3𝒙) is 3A.
10. (10%) What is the final result of A after executing the following MATLAB code?
Sigma = [1 0; 0 0.5];
S = [100 0; 0 50];
A = S*Sigma*inv(S);
(e)
a. [
1 0
].
0 1
b. [
0 0.5
].
0.5 0
c. [
100
0
0
].
100
d. [
100
0
0
].
50
e. None of the shown matrices.
S
(End of the problem sheet)