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Hi

let me know if you have any question

I'll give you a good tip if I get A on this exam!

thanks :)

M 221 — Introduction to Linear Algebra — Fall 2020
Final Exam
Due Tuesday, November 24, 2020, 11:59pm on Gradescope
Closed book, show your work, write neatly. Good luck!
Student Name:
Question:
1
2
3
4
5
6
7
8
9
Total
Points:
5
4
6
10
7
5
3
4
6
50
Score:
1. Read carefully:
(a) (1 point) Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector
is easy.
true
false
(b) (1 point) If, for some vector x, we have Ax = λx, then λ is an eigenvalue of A. (trick question!)
true
false
(c) (1 point) Eigenvectors corresponding to distinct eigenvalues for a real symmetric matrix are always orthogonal.
true
false
−2
v= 6
1
(d) (1 point) The vector v is in the null space of A:
true
A=
3 1 0
.
−5 1 −4
false
(e) (1 point) A positive definite matrix is always invertible.
true
false
1
2. (4 points) Suppose that a 2 × 2 matrix A has an eigenvalue 3 with corresponding eigenvector
and an eigenvalue −1
2
1
with corresponding eigenvector
. Find an invertible V and a diagonal D, such that A = V DV −1 :
1
A=
·
·
Page 1/6
1 −2 0
1
3. Consider the matrix B = −2 2 1 and the vector d = 0.
0
1 4
0
(a) (2 points) Compute the B = LU decomposition. (verify that B = LU !)
L=
U=
(b) (1 point) Compute the determinant det(B). (Hint: not much work needed, now...)
det(B) =
(c) (3 points) Solve Bx = d using forward/backward substitution.
x=
Page 2/6
−1 2 1
1
and b =
.
4. Consider the matrix A =
−2 4 2
2
(a) (2 points) Perform elimination on A | b to get the reduced row echelon form R | c .
R=
c=
(b) (2 points) Find the dimension of and give a basis for the column space C(A)
dim(C(A)) =
basis of C(A) =
(c) (2 points) Find the dimension of and give a basis for the null space N(A)
dim(N(A)) =
basis of N(A) =
(d) (2 points) Find the dimension of and give a basis for the row space C(AT )
dim(C(A> )) =
basis of C(A> ) =
(e) (2 points) Give the complete solution of Ax = b.
x∈
Page 3/6
0 0 1
5. Consider the matrix C = 0 1 0.
1 0 2
(a) (1 point) Compute det(C).
det(C) =
(b) (1 point) “The matrix C is invertible.”
true
false
(c) (1 point) λ1 = 1 is an eigenvalue of C. Find the associated eigenvector x1 .
x1 =
(d) (2 points) Find the other two eigenvalues. (Hint: find the characteristic polynomial and don’t simplify—polynomial
division will be easy)
λ2 =
(e) (2 points) Find A−1 by Gauss-Jordan elimination
A−1 =
Page 4/6
λ3 =
1
−1
6. (5 points) (QR decomposition) Consider the matrix A =
1
−1
1
0
.
1
0
Find matrices Q (4 × 2) with orthonormal columns and R (2 × 2) upper triangular, such that A = QR.
Q=
1
2
−1 1
7. (3 points) Let D =
0
1
1 −1
0
0
2
0
R=
2
2
. Compute det(D).
1
3
det(D) =
8. Linear independence.
(a) (2 points) Complete the definition: “A set of n vectors v1 , v2 , . . . , vn is linearly independent if and only if . . .”
Page 5/6
4
2
1
(b) (2 points) The set 2 , 5 , 1 is linearly
3
6
0
dependent
independent
Show your work:
2 4
9. Consider the matrix G =
1 2
(a) (2 points) Find eigenvalues λ1 > λ2 of G> G and their associated unit eigenvectors v1 and v2
λ1 =
λ2 =
v1 =
v2 =
(b) (2 points) Complete the SVD (e.g. using Avi = σi ui ):
G = UΣV > =
·
·
(c) (2 points) From its SVD, write down orthonormal bases for the four fundamental subspaces of G (give numbers!):
basis C(G) =
basis N(G> ) =
basis C(G> ) =
Page 6/6
basis N(G) =

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