Math Questions
imour33
HW4.pdf
Math 2568 Autumn 2020
Homework 4
For each of the following sets of vectors S,
• Find a minimal spanning set T for the subspace W = Span(S) from the set of vectors in S (T ⊆ S).
• For each vector vi ∈ S omitted from T , write it explicitly as a linear combination of the vectors in T .
Problem 1 S = {v1, v2, v3, v4, v5, v6} where
v1 =
2 3 −1 4 6
, v2 =
9 −4 −11
7 5
, v3 =
−30 25 −47 −16 −2
, v4 =
0 −5 7 11 17
, v5 =
2 0 6 11 −7
, v6 =
35 −6 11 −50 −34
,
Problem 2 S = {v1, v2, v3, v4} where
v1 =
10
1
, v2 =
−20
2
, v3 =
11
2
, v4 =
−23
1
,
Problem 3 S = {v1, v2, v3, v4, v5} where
v1 =
2 3 −1 9 5
, v2 =
6 4 5 −3 7
, v3 =
−2 2 −7 21 3
, v4 =
18 17 7 21 29
, v5 =
9 6 3 −2 1
,
Problem 4 S = {v1, v2, v3, v4} where
v1 =
1 2 −1 3
, v2 =
−2 1 2 −1
, v3 =
−1 −1 1 −3
, v4 =
−2 2 2 0
,
For each of the following matrices, find a minimal spanning set for its
• Column space
• Row space
• Nullspace
Problem 5 A =
2 3 0 1 0 7 2 1 13 16 3 −5 −3 8 22 −1 8 −1 −11 −18
Problem 6 A =
1 2 1 0 2 5 3 −1 2 2 0 2 0 1 1 −1
Problem 7 A =
4 6 10 7 211 4 15 6 1
3 −9 −6 5 10
Problem 8 A =
−1 0 5 −14 0 −12 3 2 4 −11 16 12 7 0 −5 8 22 46 −5 −5 7 −26 −13 −32 0 7 11 −19 −11 −40
For the remaining two problems, you are to determine of if the given statement is true or false, and provide a short justification for your answer (for example, an argument showing it is true, or a counter-example if it is false).
Problem 9 If S and T are two sets of vectors in Rn and every vector in T can be expressed as a linear combination of vectors in S, then every vector in S can be expressed as a linear combination of vectors in T .
Problem 10 If A is a real n × n matrix satisfying the matrix equation A ∗ A = 0n×n, then the column space of A is a subspace of the nullspace of A.
