Homework SourseLet A be the following matrix Find all vectors that re ortho
Let A be the following matrix
Find all vectors that re orthogonal to the solution space of Ax=0
| 1 | 0 | -1 | 0 |
| 1 | 2 | 0 | 3 |
| 2 | 2 | -1 | 3 |
| 3 | 2 | -2 | 3 |
Solution
Note: If A is an m x n matrix, then the nullspace of A and the row space of A are orthogonal complements in Rn with respect to the Euclidean inner product.
Note: Elementary row operations do not change the row space.
Hence we perform elementary row operations to the given matrix. Then after getting the row reduced echelon form of the matrix, the leading nonzero rows will form a basis for the row space of the matrix. Thus we can find all the vectors which are orthogonal to Ax=0.
A
1
0
-1
0
1
2
0
3
2
2
-1
3
3
2
-2
3
1
0
-1
0
R2-R1
0
2
1
3
is equivalent to
R3-2R1
0
2
1
3
R4-3R1
0
2
1
3
1
0
-1
0
is equivalent to
0
2
1
3
R3-R2
0
0
0
0
R4-R2
0
0
0
0
Thus, first and second row are the vectors which span the row space of A. So the answer is the space spanned by
{(1, 0, -1, 0)T, (0, 2, 1, 3)T}={v1,v2}
Any vector of the form v=av1+bv2, a,b scalars.
| A | ||||||||
| 1 | 0 | -1 | 0 | |||||
| 1 | 2 | 0 | 3 | |||||
| 2 | 2 | -1 | 3 | |||||
| 3 | 2 | -2 | 3 | |||||
| 1 | 0 | -1 | 0 | |||||
| R2-R1 | 0 | 2 | 1 | 3 | ||||
| is equivalent to | R3-2R1 | 0 | 2 | 1 | 3 | |||
| R4-3R1 | 0 | 2 | 1 | 3 | ||||
| 1 | 0 | -1 | 0 | |||||
| is equivalent to | 0 | 2 | 1 | 3 | ||||
| R3-R2 | 0 | 0 | 0 | 0 | ||||
| R4-R2 | 0 | 0 | 0 | 0 | ||||
