Download MATH10212 • Linear Algebra • Examples 2 Linear dependence and

MATH10212 • Linear Algebra • Examples 2
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3, 5(b), 6, 8, 9, 11, 12;
Linear dependence
and independence
Week 3, 16/2–20/2
6*. [2.3.19] Prove that ~u, ~v , and w
~ belong to
span(~u, ~u + ~v , ~u + ~v + w).
~
(Textbook, Subsection 2.3)
In exercises 1–3 [2.3.1, 3, 5]1 , determine if the vector ~v is a linear combination of the given vectors
~ui .
¸
¸
¸
·
·
·
1
1
2
, ~u1 =
, ~u2 =
1. ~v =
2
−1
−1
 
 
 
1
1
0
2. ~v =  2 , ~u1 =  1 , ~u2 =  1 
3
0
1
 
 
 
1
0
1
3*. ~v =  2 , ~u1 =  1 , ~u2 =  1 ,
3
0
1
 
1
~u3 =  0 
1
Use the method of Example 2.23 and Theorem 2.6
from the book to determine if the sets of vectors in
Exercises 7–10 are linearly independent. If, for any
of these, the answer can be determined by inspection (i.e., without calculation), state why. For any
sets that are linearly dependent, find a dependence
relationship (coefficients) among the vectors.
    

1
1
1
7. [2.3.23]  1 ,  2 ,  −1 
1
3
2
     
2
2
0
8*. [2.3.25]  1 ,  1 ,  0 
2
3
1
     
3
6
0
9*. [2.3.27]  4 ,  7 ,  0 
5
8
0

 
 
 

1
−1
1
0
 −1   1   0   1 
 
 
 

10. [2.3.29] 
 1 ,  0 ,  1 ,  −1 
0
1
−1
1
4. [2.3.7] Determine if the vector ~b is in the span
of the columns of matrix A.
·
¸
·
¸
1 2
5
~
A=
, b=
3 4
6
5. (a) [2.3.9] Show that
¸ ·
µ·
¸¶
1
1
2
,
R = span
.
1
−1
11*. Prove that for any ~u, ~v ∈ Rn , the three vectors ~u + 2~v , ~v − ~u, 2~u + ~v are linearly dependent.
(b)* [2.3.11] Show that
12*. Use the “row method” of Theorem 2.7 to decide if the vectors [−1, 1, −2, −1], [1, −1, 1, 1],
[1, −1, −1, 1] are linearly dependent or independent (it is not required to find a dependence).
     
1
1
0
R3 = span 0 , 1 , 1 .
1
0
1
1 Exercise
numbers in square brackets refer to the text:
D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7.
The textbook contains answers to all odd numbered problems.
1