THE FINAL COVERS: CH 3, CH 4, CH 5, and 6.1, 6.2
You should know the statement of Theorem 6.6 (from 6.1) , not the proof, I might
ask true-false questions related to Theo 6.6 (see problem 5)
1.
Find the eigenvalues and the bases of the related eigenspaces for:
210
A = [ 0 2 3]
005
A=[
−1 9
]
−2 8
0 0 3
A= [1 0 − 1]
0 1 3
410
A= [3 4 1]
014
Answer are these matrices diagonalizable?
2.
a)
b)
c)
d)
e)
Which of the following sets are a basis (explain why or why not)
(1,0,1), (1,1,-1), (2, 1,0) in 𝑅 3
(2,1,-1), (1,0,1), (3,1, 2) in 𝑅 3
(2,3,1), (4,1,2) in 𝑅 3
1, x+1, 𝑥 2 +1 in 𝑃2
𝑥 2 , x+1, 𝑥 2 − 2𝑥 − 2 in 𝑃2
3.
Suppose T is a rotation by 𝝅/3 , S is a rotation by 𝝅/4 and F is reflection over the y-axis in 𝑅 2
a) Write matrices for T, S, F with respect to standard basis in 𝑅 2
b) Find the matrices for compositions T ⃙ S, S ⃙ T, compositions T ⃙ F, F ⃙ T
c) Which of these compositions are commutative?
4. For the following matrices find the bases for: nullspace, rowspace, columnspace
Rank and nullity:
1 2 01
A = [−1 0 1 1]
1 4 13
1 2 0 1
1 0 1 1
A=[
]
0 2 −1 1
2 4 0 3
:
5. Suppose A is a nxn matrix. True or false:
a)
b)
c)
d)
e)
f)
If detA ≠0, then Ax=b has a unique solution for every real b
If detA ≠0, then the rows of A are linearly independent
If A is invertible then det A=0
The rows of A are linearly independent if the columns of A are linearly independent
If A has nullity 0, then detA ≠0
If A has nullity > 0, then detA ≠0
6. Suppose W is the span of the vectors (1,2,0,1), (-1,0,1,1), (1,4,1,3). Find the basis of 𝑊 ⊥ .
(hint: use A from problem 4.)
7. Apply the Gram Schmidt process to the vectors (-1,2,1), (0,-1,1), (3,3,1).
8. Suppose that V is an inner product space, u,v in V. Show:
a) II 𝑢 + 𝑣 II 2 + II 𝑢 − 𝑣 II 2 = 2( II 𝑢 II 2 + II 𝑣 II 2 )
b) If u,v are orthonormal then II 𝑢 + 𝑣 II 2 = II 𝑢 − 𝑣 II 2 = 2