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HW 9

This document is a homework assignment for a Linear Algebra course due on December 3, 2024. It includes various problems related to linear dependence of vectors, spans, null spaces, and finding bases for different vector spaces. The problems require students to analyze sets of vectors in R3 and perform calculations involving linear combinations and matrix properties.

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wiam Hmamouche
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21 views2 pages

HW 9

This document is a homework assignment for a Linear Algebra course due on December 3, 2024. It includes various problems related to linear dependence of vectors, spans, null spaces, and finding bases for different vector spaces. The problems require students to analyze sets of vectors in R3 and perform calculations involving linear combinations and matrix properties.

Uploaded by

wiam Hmamouche
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Linear Algebra HW 9 Fall 2024

Due: December 3, 2024

1. Which of the following sets of vectors in R3 are linearly dependent? If it is linearly dependent,
write one vector as a linear combination of the other vectors.
   
 1 3 
(a)  2 , 2
 
−1 5
 
     
 4 2 1 
(b)  2 , 6 , −2
   
1 −5 3
 
       
 1 0 1 3 
(c) 1 , 2 , 2 , 6
0 3 3 6
 
      
 1 1 1 
(d)  2 , 1 , 0
   
3 1 1
 

2. For what values of c ∈ R3 are the vectors


     
−1 2 1
 0  , 1  ,  1
−1 2 c

linearly dependent?

3. Let ~v1 , ~v2 , ~v3 be independent vectors in a vector space V . Define new vectors

~ 1 = ~v1 + ~v2
w ~ 2 = ~v2 + ~v3
w ~ 3 = ~v1 + ~v3 .
w

Are the vectors w


~ 1, w ~ 3 independent ? Show work.
~ 2, w

4. Take the vectors        


1 a 1 3
~v1 = 1 , ~v2 = 2a , ~v3 = 1 and ~u = 4 .
      
1 1 b 4
Find all values a, b ∈ R such that

(a) ~u 6∈ Span(~v1 , ~v2 , ~v3 )


(b) ~u is a unique linear combination of the vectors {~v1 , ~v2 , ~v3 }
(c) ~u ∈ Span(~v1 , ~v2 , ~v3 ) but is not a unique linear combination of the vectors {~v1 , ~v2 , ~v3 }
     
1 −1 2
−2  2  −4 4
5. Describe the subspace S = Span   3  , −2 ,  5  in R as a null space. This means
     

2 −1 3
find a matrix B such that S = N (B) = null space of B
 
1 1 1 0
6. Find a basis for the nullspace of the matrix A = .
2 1 0 1
7. For the following matrix  
1 2 −1 1
A = 2 4 −3 0
1 2 1 5

(a) find a basis for the null space


(b) find a basis for the column space using the transpose matrix AT .
(c) find a basis for the row space which is a subset of the original row vectors.

8. For each matrix below, find a basis for the row space and the column space:
 
  −2 2 3 7 1
1 −4 9 −7 −2 2 4 8 0
A = −1 2 −4 1  , B= −3

3 2 8 4
5 −6 10 7
4 −2 1 −5 −7
         
1 0 1 1 −1
1 1  0 1 −5
 0  , ~v2 = 2 , ~v3 =  1  , ~v4 = −6 , ~v5 =  1 . What is the dimension of
9. Let ~v1 =          

−1 1 −1 −3 0
S = Span(~v1 , ~v2 , ~v3 , ~v4 , ~v5 )? Find a basis.

10. In each case below, find the dimension of the subspace:


  

 a + c 

−a + b
  
4
(a) S =    ∈ R : a, b, c ∈ R

 −b − c  

a + b + 2c
 
  
a11 a12
(b) S = : 3a11 − 5a21 − a12 = 0, a21 + 4a11 − a22 = 0
a21 a22
(c) S = {A ∈ M3 (R) : A is symmetric }

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