Math 317: Linear Algebra
Homework 10
Due: TBA
The following problems are for additional practice and are not to be turned in: (All
problems come from Linear Algebra: A Geometric Approach, 2nd Edition by ShifrinAdams.)
Exercises: Section 4.3: 5a, 6, 7, 11, 13, 16, 17 Section 5.1 : 1, 3, 10, 11
Turn in the following problems.
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1. Continuing the example from class, let V = span {v1 , v2 } where v1 = 0 , v2 =
1
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 1 . Recall that we found a basis for R3 by taking v3 = −2 ∈ V ⊥ .
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(a) Find the standard matrix of the linear transformation T : R3 → R3 , if T (x) =
projV ⊥ (x). That is, calculate [projV ⊥ ]stand . Hint: Recall that for any x ∈ R3 ,
we have that x = projV (x) + projV ⊥ (x). Use the fact that Ix = x where I
is the identity transformation and your knowledge of [projV ]stand to calculate
[projV ⊥ ]stand .
(b) Calculate the matrix of the linear transformation T : R3 → R3 , if T (x) =
projV ⊥ (x) with respect to B, if B = {v1 , v2 , v3 }. That is, calculate [projV ⊥ ]B .
2. Section 4.3, Problem 5b
3. Section 4.3, Problem 9
4. Section 4.3, Problem 18
5. Section 4.3, Problem 19
6. Section 4.3, Problem 20
7. Prove that the rank and trace of a matrix are similarity invariants. That is
rank(An×n ) = rank(Bn×n ) and trace(An×n ) = trace(Bn×n ) when A is similar to
B.
8. Section 5.1, Problem 1a
9. Section 5.1, Problem 2
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