Question 1: Using the reduced row echelon form method determine whether followin
ID: 3099548 • Letter: Q
Question
Question 1:
Using the reduced row echelon form method determine whether following set of vectors S={v1,v2,v3,v4} in R4 is linearly independent or linearly dependent? v1=(1,0,-1,0),v2=(0,2,0,-2),v3=(0,0,3,0),v4=(-4,0,4,0)
Question 2:
Determine whether or not the set S={v1,v2} spans the vector space R2, where v1=(1,1),v2=(0,1)?
Question 3:
Determine whether or not the linear operator T: R2 to R2 defined by the equation
w1=4x1-6x2
w2=-2x1+3x2
is one-to-one; if so, find the standard matrix for the inverse operator and find T-1(w1,w2)
Explanation / Answer
Write to a matrix:
[v1 v2 v3 v4] =
1
0
0
-4
0
2
0
0
-1
0
3
4
0
-2
0
0
Add the first row to the third makes:
1
0
0
-4
0
2
0
0
0
0
3
0
0
-2
0
0
Add the second row to the last row makes:
1
0
0
-4
0
2
0
0
0
0
3
0
0
0
0
0
The matrix above is the matrix in row reduced echelon form
Since there are only 3 pivots ( in row 1 it's 1, in row 2 it's 2, in row 3 it's 3), the vectors are linearly dependent!
The first row tells us that the first and 4th vector are linearly dependent. Apparently v1 = -4*v4
And indeed:
1 * -4 = -4
0 * -4 = 0
-1 * -4 = 4
0 * -4 = 0
1
0
0
-4
0
2
0
0
-1
0
3
4
0
-2
0
0
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