linear algebra
Spanning, Linear Independence, and Bases
July
1
4,
2
02
0
Question 1
a)Is the vector
(
2
4
)
a linear combination of the vectors
( (
1
1
)
,
(
1
−3
) )
b)Is the vector
28
11
a linear combination of the vectors (
14
0
,
18
6
,
−1−12
−1
)
c) Is the vector
1−1
−8
a linear combination of the vectors (
10
0
,
11
4
,
0−1
−4
)
Question 2
Answer the Challenge question from HW 1. (You can now answer it quickly,
once you rephrase it in terms of spanning vectors)
Question 3
For the following, give an example if one exists, or state it is not possible. If
it is not possible, explain why.
a) A sequence of 3 vectors in R3 that are LI.
b) A sequence of 2 vectors in R3 that are spanning vectors.
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c) A sequence of 4 vectors in R2 that are spanning vectors.
d) A sequence of 3 vectors that are a basis for R3
e) A sequence of 3 vectors that are a basis for R4
Question 4
State whether the following are true or false. For those that are false, give a
counter example.
a) Every list of two vectors in R3 are LI.
b) Every list of four vectors in R3 are spanning.
c) Every list of four linearly independent vectors in R4 are a basis.
d) Every list of five vectors in R5 that span R5 are a basis.
2