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Quiz Solutions for MATH 304 - Vector Operations, Quizzes of Linear Algebra

Western Washington University (WWU)Linear Algebra

The solutions to quiz 2 of math 304, focusing on vector operations. Topics covered include finding the length of vectors, determining if one vector is shorter than another, finding unit vectors, projections, and orthogonal vectors.

Typology: Quizzes

Pre 2010

Uploaded on 08/19/2009

koofers-user-1f6-1 🇺🇸

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MATH 304 Quiz 2 Solutions

February 6, 2007 Name

Put your answers in the space provided. Show your reasoning. Calculators may not be used. The maximum

score on this quiz is 6 points.

1. 4 points Consider the following two vectors in R3:u= [−2,1,3]Tand v= [3,3,4]T.

1a. Prove that uis shorter than v.

||u||2= 4 + 1 + 9 = 14,||v||2= 9 + 9 + 16 = 34. Therefore uis shorter than v

1b. Compute the distance from uto v

The distance from uto vis ||v−u|| =p(3 −(−2))2+ (3 −1)2+ (4 −3)2=√25+4+1=√30

1c. Find an unit vector in the direction opposite to u

From 1a. the length of uis √14. The answer is −1

√14 [−2,1,3]T

1d. Find the projection of uonto v.

The projection is u·v

v·vv=−6+3+12

9+9+16 v=9

34 [3,3,4]T

1e. Find a non-zero vector which is orthogonal to u

Two possible answers are [−1,1,−1]Tor [0,3,−1]T

1f. Find uvT

uvT=





−2

1

3





[3,3,4] = 





−6−6−8

334

9 9 12







2. 2 points Let Wbe the subspace of R4spanned by u1= [1,1,0,0]Tand u2= [1,−1,0,0]T. Find a basis

for W⊥. Circle your answer.

There are many answers. One is

















0

1







,







0

1

2

















Discover Quizzes of Linear Algebra Western Washington University (WWU)

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Download Quiz Solutions for MATH 304 - Vector Operations and more Quizzes Linear Algebra in PDF only on Docsity!

MATH 304

Quiz 2 Solutions February 6, 2007 Name

Put your answers in the space provided. Show your reasoning. Calculators may not be used. The maximum score on this quiz is 6 points.

4 points Consider the following two vectors in R^3 : u = [− 2 , 1 , 3]T^ and v = [3, 3 , 4]T^.

1a. Prove that u is shorter than v.

||u||^2 = 4 + 1 + 9 = 14, ||v||^2 = 9 + 9 + 16 = 34. Therefore u is shorter than v

1b. Compute the distance from u to v

The distance from u to v is ||v−u|| =

√ (3 − (−2))^2 + (3 − 1)^2 + (4 − 3)^2 =

1c. Find an unit vector in the direction opposite to u

From 1a. the length of u is

The answer is

[− 2 , 1 , 3]T

1d. Find the projection of u onto v.

The projection is

u · v v · v

v =

[3, 3 , 4]T

1e. Find a non-zero vector which is orthogonal to u

Two possible answers are [− 1 , 1 , −1]T^ or [0, 3 , −1]T

1f. Find uvT

uvT^ =

  

   [3,^3 ,^ 4] =

  

  

2 points Let W be the subspace of R^4 spanned by u 1 = [1, 1 , 0 , 0]T^ and u 2 = [1, − 1 , 0 , 0]T^. Find a basis

for W ⊥. Circle your answer.

There are many answers. One is

    

   

   