Problem 1. You\'ll want a calculator for this problem. Consider the vectors 70 p
ID: 3117112 • Letter: P
Question
Problem 1. You'll want a calculator for this problem. Consider the vectors 70 pts. 25 U2 U5 13 26 0 0 Let S span(U1, U2, U3,U4), which is a subspace of R A. Find a basis of S. What is the dimension of S? B. Express the vectors in the list vi, , v4 that are not part of the basis you found as linear combinations of the basis vectors C. Consider the vectors 70 10 81 w1 = 47 Determine if these vectors are in S. If the vector is in S, express it as a linear combination of the basis vectors found aboveExplanation / Answer
Let A = [v1,v2,v3,v4,v5,w1,w2] =
5
5
-5
5
25
70
8
1
0
-2
3
4
10
-2
5
13
3
-11
26
81
47
0
1
1
-2
0
1
5
To answer the questions, we will reduce A to its RREF as under:
Multiply the 1st row by 1/5
Add -1 times the 1st row to the 2nd row
Add -5 times the 1st row to the 3rd row
Multiply the 2nd row by -1
Add -8 times the 2nd row to the 3rd row
Add -1 times the 2nd row to the 4th row
Multiply the 3rd row by -1/7
Add 1 times the 3rd row to the 4th row
Multiply the 4th row by -35/2
Add 51/35 times the 4th row to the 3rd row
Add -18/5 times the 4th row to the 2nd row
Add -8/5 times the 4th row to the 1st row
Add -1 times the 3rd row to the 2nd row
Add -5 times the 3rd row to the 1st row
Add -1 times the 2nd row to the 1st row
Then the RREF of A is
1
0
-2
3
0
-2
0
0
1
1
-2
0
1
0
0
0
0
0
1
3
0
0
0
0
0
0
0
1
A. Now it is apparent that a basis for S is {v1,v2 } = {(5,1,5,0)T,(5,0,13,1)T} or {(1,0,0,0)T,(0,1,0,0)T}.The dimension of S is 2.
B. v3 = -2v1+v2 and v4 = 3v1-2v2.
C. None of w1 and w2 is in S.
Note:
There appears to be a misprint here. Possibly, S = span(v1,v2,v3,v4,v5) and NOT (v1,v2,v3,v4).
Then the answer change as under:
A.a basis for S is {v1,v2,v5 } = {(5,1,5,0)T,(5,0,13,1)T,(25,4,26,0)T} or {(1,0,0,0)T,(0,1,0,0)T, (0,0,1,0)T}.The dimension of S is 3.
B. v3 = -2v1+v2 and v4 = 3v1-2v2.
C. w1 =-2v1+v2+3v5. Further, w2 is not in S.
5
5
-5
5
25
70
8
1
0
-2
3
4
10
-2
5
13
3
-11
26
81
47
0
1
1
-2
0
1
5
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