Name:Due Date: Thurs. Dec. 7th

In Class Peer Review Assignment 3

D.M. 1 : 9 (9pts) Short Answer 10 : 16 (12pts) Proofs 17 : 18 (17pts) T/F 19 : 30 (11pts)

Total out of (49pts)

Directions: Put only your answers on this assignment. Do all scratch work and side calcula-tion on a separate piece of scratch paper. If you need to attach work please staple it on the backof this assignment. Write neatly and legibly. If I can’t read it you will receive a zero for thatproblem. Finally be sure to show all of your work (justification) for each problem to receive fullcredit.

Grader Names:

1

Table 1: Some key terms from exam 2 material.

Characteristic Equation Similar Matrices Orthogonal Vectors Dot ProductUnit Vector Distance Orthogonal Complement Orthogonal Set

Orthogonal Projection Normalized Orthonormal Set Orthonormal MatrixGram Schmidt Orthonormal Basis Orthogonal Basis Length/Magnitude/L2 Norm

Definition Matching: (1pt each) Choose the most appropriate key term in Table 1.

1. If v ∈ Rn then we define√·v to be the of v.

2. If v,w ∈ R! then we define ||v−w|| to be the

between v and w.

3. If v ∈ Rn such that ||v|| = 1 then v is a

.

4. If v,w ∈ R! such that v ·w = 0 then v,w are said to be

.

5. If A,B ∈ Rn×n and P is an invertible matrix s.t. A = PBP−1 then we say A and B are

.

6. Given a subspace W of Rm with a basis B = {b1, b2, . . . bn} we define projb1y+ projb2

y+ · · ·+ projbny as

the of a vector y ∈ Rn onto the subspace W.

7. Given a subspace W of Rm with a basis B = {b1, b2, . . . bn} we define the set {v − 1 = b1, v2 = b2 −

projv1b2, v3 = b3 − projv1

b3 − projv2b3, . . . , vn = bn −

n−1!i=1

projvibn} as a

of the subspace W.

8. The process used to find the set in question 7 is known as themethod.

9. An orthogonal set whose vectors all have magnitude 1 is known as a

.

2

Short Answer: (2pts each)

Let x1 =

⎡

⎢⎢⎢⎢⎣

01

−121

⎤

⎥⎥⎥⎥⎦, x2 =

⎡

⎢⎢⎢⎢⎣

10101

⎤

⎥⎥⎥⎥⎦, B = {x1,x2}, W = Span{B}, z =

⎡

⎢⎢⎢⎢⎣

1321

−3

⎤

⎥⎥⎥⎥⎦, y =

⎡

⎢⎢⎢⎢⎣

01131

⎤

⎥⎥⎥⎥⎦.

10. Show that B is not an orthogonal set of vectors.

11. Explain why B is a basis for W.

12. Show that z ∈ W⊥.

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13. Find an orthonormal basis for W.

14. Find vectors y and n such that y ∈ W, n ∈ W⊥, and y = y + n.

4

15. Find a nonzero vector in R5 that is othogonal to both

⎡

⎢⎢⎢⎢⎣

11111

⎤

⎥⎥⎥⎥⎦and

⎡

⎢⎢⎢⎢⎣

10204

⎤

⎥⎥⎥⎥⎦.

16. Find the orthogonal projection of

⎡

⎢⎢⎢⎢⎢⎢⎣

100000

⎤

⎥⎥⎥⎥⎥⎥⎦onto

⎡

⎢⎢⎢⎢⎢⎢⎣

111111

⎤

⎥⎥⎥⎥⎥⎥⎦.

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Proofs :

17. State the Invertible Matrix Theorem with all the components we have covered. (5pts)

6

18. Prove equivalency between 4 of the properties in the Invertible Matrix Theorem of your choice. (12pts)

7

T/F: (1pt each)

19. : If λ+ 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.

20. : v · v = ||v||2 for any vector v ∈ Rn

21. : For any scalar c, u · (cv) = c(u · v).

22. : For a square matrix A, any vector in the row space of A is orthogonal to any vector in the Null(A).

23. : If B is orthogonal basis for a subspace W and c = 0 is a scalar then multiplying a vector of B by cwill result in a new orthogonal basis.

24. :u · v− v · u = 0 for any vectors u, v ∈ Rn.

25. : For any scalar c, ||cv|| = c||v||

26. : If B is a basis for a subspace W and x is orthogonal to every vector of B then x ∈ W⊥

27. : If W a subspace of Rn and v is a nonzero vector in W, then it is possible that v can also be inW⊥.

28. : If ATx = 0, then x ⊥ Col(A).

29. : If W is any subspace of R3, then dim(W) + dim(W⊥) = 3.

30. : It is possible for u to not be orthogonal to v and not be orthogonal to w, but be orthogonal to(v +w)

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