For each matrix in problem 4, determine if the columns of the matrix span R3. Gi
ID: 3036116 • Letter: F
Question
For each matrix in problem 4, determine if the columns of the matrix span R3. Give reasons for your answers. (Make as few calculations as possible.)
Matrices are : it is hard to use row and column in here but, i will do the following.
Matrix A = [column 1 column 2] , where column 1 = (-3 2 4)
Column 2= (12 6 8)
Matrix B has three columns;[ column 1 column 2 column 3] where
Column 1 contains (3 -2 6)
Column 2 contains (7 -5 11)
Column 3 contains (1 3 -4)
where columns are vertical position in matrix.
Matrix C has four columns;
Column 1 ( -2 4 0)
Column 2 (3 0 0)
Column 3 (5 -5 0)
Column 4 (1 11 0)
Explanation / Answer
1.We have A =
-3
12
2
6
4
8
We will reduce A to its RREF as under:
Multiply the 1st row by -1/3 ; Add -2 times the 1st row to the 2nd row
Add -4 times the 1st row to the 3rd row ; Multiply the 2nd row by 1/14
Add -24 times the 2nd row to the 3rd row; Add 4 times the 2nd row to the 1st row
Then theRREF of A is
1
0
0
1
0
0
Since neither of the columns in the RREF of A has 1 in the last row, the columns of A do not span R3.
2 B =
3
7
1
-2
-5
3
6
11
-4
We will reduce B to its RREF as under:
Multiply the 1st row by 1/3; Add 2 times the 1st row to the 2nd row
Add -6 times the 1st row to the 3rd row; Multiply the 2nd row by -3
Add 3 times the 2nd row to the 3rd row ; Multiply the 3rd row by -1/39
Add 11 times the 3rd row to the 2nd row ; Add -1/3 times the 3rd row to the 1st row
Add -7/3 times the 2nd row to the 1st row
Then the RREF of B is I3. Hence the columns of B span R3.
3. C =
-2
3
5
1
4
0
-5
11
0
0
0
0
We will reduce C to its RREF as under:
Multiply the 1st row by -1/2 ; Add -4 times the 1st row to the 2nd row
Multiply the 2nd row by 1/6 ; Add 3/2 times the 2nd row to the 1st row
Then the RREF of C is
1
0
-5/4
11/4
0
1
5/6
13/6
0
0
0
0
Since none of the columns in the RREF of C has 1 in the last row, the columns of C do not span R3.
-3
12
2
6
4
8
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