Determine if the columns of the matrix span R4 15 9 -12 15 -610 2-3 21 -27 -6 33
ID: 3169427 • Letter: D
Question
Determine if the columns of the matrix span R4 15 9 -12 15 -610 2-3 21 -27 -6 33 Select the correct choice below and fill in the answer box to complete your choice A. The columns span R4 because at least of the columns of A is a linear combination of the other columns of A B. The columns do not span R4 because none of the columns of A are linear combinations of the other columns of OC. The columns span R4 because the reduced row echelon form of the augmented matrix iswhich has a pivot in every row. (Type an integer or decimal for each matrix element.) OD. The columns do not span R4 because the reduced row echelon form of the augmented matrix is, which does not have a pivot in every row Type an integer or decimal for each matrix element.)Explanation / Answer
Let the given matrix be denoted by A. To detrermine whether the columns of A span R4, we will reduce A to its RREF as under:
Multiply the 1st row by -1/7
Add -15 times the 1st row to the 2nd row
Add 6 times the 1st row to the 3rd row
Add -21 times the 1st row to the 4th row
Multiply the 2nd row by 7/33
Add 58/7 times the 2nd row to the 3rd row
Add 33 times the 2nd row to the 4th row
Multiply the 3rd row by -11/50
Add 3/11 times the 3rd row to the 2nd row
Add 5/7 times the 3rd row to the 1st row
Add -2/7 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
-73/50
0
1
0
33/50
0
0
1
-129/50
0
0
0
0
This implies that the columns of A do not span R4 as the last column of A is a linear combination of its first 3 columns. Thus the dimension of col(A) is 3 while the dimensuion of R4 is 4. The option B is partially correct in the sense that “the columns of A do not span R4”. However, the part “because none of the columns of A are linear combinations of the other columns of A “ is INCORRECT.
1
0
0
-73/50
0
1
0
33/50
0
0
1
-129/50
0
0
0
0
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