Linear Algebra Final 2019 Fall N ame :
1. a. Let a vector space be spanned by r linearly independent vectors. What is the dimension of the vector space?
b. Can an orthogonal set of vectors be linearly dependent?
c. Write the four fundamental spaces of the matrices.
2. Let the matrix A have the echelon form U.
2 −1 −1 2 −1 2 −1 −1 2 −1
Find the linearly independent vectors −2 1 4 −3 3 Echelon 0 0 3 −1 2
spanning the row space of A A= −−−−−−→ U =
−4 2 −1 −3 2 0 0 0 0 2
Find the linearly independent vectors
−6 3 −3 −4 −3 0 0 0 0 0
spanning the column space of A
Row space = α [ 2 -1 -1 2 -1 ]T + β [ 0 0 3 -1 2 ]T + γ [ 0 0 0 0 2 ]T
Column space = α [ 2 -2 -4 -6 ]T + β [ -1 4 -1 -3 ]T + γ [ -1 3 2 -3 ]T
3. Let the matrix A have the reduced row
echelon form R. 1 −2 −2 −3 −3 Reduced 1 −2 0 1 −1
−2 4 5 8 7 Echelon 0 0 1 2 1
Let x = [x1 , x2 , x3 , x4 , x5 ]T be the A=
1 −2
−−−−−−→ R =
solution for the matrix equation Ax = 0. 0 1 −1 0 0 0 0 0
2 −4 −3 −4 −5 0 0 0 0 0
Hint: Ax = 0 ⇔ Rx = 0.
a) Solve the matrix equation Ax = 0. b) Find the null space of the matrix A.
x1 x5 − x4 + 2x2 2 −1 1
x2 x2 1 0 0
x3 = −x5 − 2x4 null(A) = x2 0 + x4 −2 + x5 −1
x x4 0 1 0
4
x5 x5 0 0 1
4. The vectors u1 = [ 1 1 ]T and u2 = [ 1 − 1 ]T are orthoganal. Find y = [ 6 − 8 ]T as their linear combination.
Hint: Express y = α1 u1 + α2 u2 where the coefficients αi = y · ui /(ui · ui ).
1 1
−2
y · u1 = [ 6 − 8 ] 1 = −2 u1 · u1 = [ 1 1 ] 1 =2 α1 = 2 = −1
1 1
−1 −1 14
y · u2 = [ 6 − 8 ] = 14 u2 · u2 = [ 1 − 1 ] =2 α2 = 2 =7
y = α1 u1 + α2 u2 = −u1 + 7u2 .
5. Find the eigen values of
the following matrices. 1 0 0 1 4 6
1 2
M1 = , M2 = 4 2 0 , M3 = 0 2 5 .
2 4
6 5 3 0 0 3
Characteristic 1−λ 2
= (1 − λ)(4 − λ) − 4 Eigen values
polynomial 2 4−λ ⇒ λ = 0, λ2 = 5.
are the roots 1
for M1 = λ2 − 5λ
Characteristic
Eigen values λ1 = 1, λ2 = 2,
polynomial of |M2 | = (1 − λ)(2 − λ)(3 − λ) ⇒
are the roots λ3 = 3.
triangular M2
Characteristic
Eigen values λ1 = 1, λ2 = 2,
polynomial of |M3 | = (1 − λ)(2 − λ)(3 − λ) ⇒
are the roots λ3 = 3.
triangular M3
1
6. Let the matrix A have the following eigen decomposition.
−1
1 2 1 0 1/2 3/5 −1 0 0 0 1/2 3/5
A = −2 1 1 = −1/2 −1/2 −1/5 0 1 0 −1/2 −1/2 −1/5
4 2 0 1 1 1 0 0 2 1 1 1
a. Tell the eigenvalues and eigenvectors of the A b. Write Ak in terms of eigen values and vectors.
1st eigen value and vector 2nd eigen value and vector 3rd eigen value and vector
0 1/2 3/5
λ1 = −1 , v1 = − 1/2
λ2 = 1 , v2 = − 1/2
λ3 = 2 , v 3 = − 1/5
1 1 1
k −1
1 2 1 0 1/2 3/5 (−1)k 0 0 0 1/2 3/5
Ak =
−2 1 1 = −1/2 −1/2 −1/5
0 1k −1/2 −1/2 −1/5
0
4 2 0 1 1 1 0 0 2k 1 1 1
h 2 −1 −1 i
7. The matrix A = 4 2 4 has the eigenvalues λ1 = 3, λ2 = 4, λ3 = −2.
−2 3 1
a) Find the eigen vector for λ1 = 3 b) Find the eigen vector for λ2 = 4 c) Find the eigen vector for λ3 = −2
−1 −1 −1 −2 −1 −1 4 −1 −1
A − λ1 I =
4 −1 4
A − λ2 I =
4 −2 4
A − λ3 I =
4 4 4
−2 3 −2 −2 3 −3 −2 3 3
Gaussian −1 −1 −1 Gaussian −2 −1 −1 Gaussian 4 −1 −1
eliminate eliminate eliminate
1st
=
0 −5 0
1st
=
0 −4 2
1st
= 0
5 5
column column column
0 5 0 0 4 −2 0 5/2 5/2
Gaussian −1 −1 −1 Gaussian −2 −1 −1 Gaussian 4 −1 −1
eliminate eliminate eliminate
2nd
=
0 −5 0
2nd
=
0 −4 2
2nd
= 0
5 5
column column column
0 0 0 0 0 0 0 0 0
| {z } | {z } | {z }
U matrix U matrix U matrix
Reduced
1 0 1 Reduced
1 0 3/4 Reduced
1 0 0
Row
Row
Row
Echelon =
0 1 0
Echelon =
0 1 −1/2
Echelon =
0 1 1
F orm F orm F orm
0 0 0 0 0 0 0 0 0
| {z } | {z } | {z }
R matrix R matrix R matrix
(A − λ1 I)x = 0 ⇔ Rx = 0 (A − λ2 I)x = 0 ⇔ Rx = 0 (A − λ3 I)x = 0 ⇔ Rx = 0
1 0 1 x1 1 0 3/4 x1 1 0 0 x1
0 1 x2 = 0
0 x2 = 0 x2 = 0
0 1 −1/2 0 1 1
0 0 0 x3 0 0 0 x3 0 0 0 x3
x1 + x3 = 0 x1 + (3/4)x3 = 0 x1 = 0
x2 = 0 x2 + (−1/2)x3 = 0 x2 + x3 = 0
x1 −1 x1 − 3/4 x1 0
x2 = x3 0 x2 = x3 1/2 x2 = x3 −1
x3 1 x3 1 x3 1
