Let A be the matrix (a) Find a basis of the column space. Find the coordinates o
ID: 3136661 • Letter: L
Question
Let A be the matrix
(a) Find a basis of the column space. Find the coordinates of the dependent
columns relative to this basis.
(b) What is the rank of A?
(c) Use the calculations in part (a) to find a basis for the row space.
3. Find a different basis of the row space of the matrix A in the previous question
using the Gauss-Jordan elimination on A^T . Find the coordinates of the
dependent rows relative to this basis.
4. Let A be the matrix from question 2.
(a) Find a basis of the homogeneous space of A.
(b) What is the dimension of the homogeneous space of A?
(c) Let n be the number of columns in A. Verify that the dimension of the
homogeneous space is equal to n - rank.
I just need question 3 &4
5121 4133 3145 1012Explanation / Answer
For answering the questions, we have to reduce A to its RREF, which can be done as under:
Add -1 times the 1st row to the 3rd row
Add -2 times the 1st row to the 4th row
Add 7 times the 2nd row to the 3rd row
Add 11 times the 2nd row to the 4th row
Add -3 times the 2nd row to the 1st row
Then the RREF of A is
1
0
1
2
0
1
1
1
0
0
0
0
0
0
0
0
3. Thus { (1,3,4,5),(0,1,1,1)} and {(1,0,12),(0,1,1,1)} are bases for Row (A).
We have AT =
1
0
1
2
3
1
-4
-5
4
1
-3
-3
5
1
-2
-1
This matrix can be reduced to its RREF as under:
Add -3 times the 1st row to the 2nd row
Add -4 times the 1st row to the 3rd row
Add -5 times the 1st row to the 4th row
Add -1 times the 2nd row to the 3rd row
Add -1 times the 2nd row to the 4th row
Then the RREF of AT is
1
0
1
2
0
1
-7
-11
0
0
0
0
0
0
0
0
Thus, the 1st and the 2nd columns of AT i.e. the 1st and the 2nd rows of A are linearly independent. The 3rd and the 4th columns of AT i.e. the 3rd and the 4th rows of A are dependent and are linear combinations of the first 2 rows of A. Further, (1,-4,-3,-2) = 1(1,0,12) -7(0,1,1,1) = 1R1 -7R2.
4.Please post question 4 again separately.
1
0
1
2
0
1
1
1
0
0
0
0
0
0
0
0
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