Homework SourseLet W be the subspace of R4 spanned by 1 2 3 4 5 6 7 8 9 10
Solution
Let X = (x,y,z,w) be an arbitrary vector in W. Then (x,y,z,w)T.( 1,2,3,4) = 0 or, x+2y+3z+4w = 0…(1). Similarly, 5x+6y+7z+8w = 0…(2), 9x+10y +11z +12w = 0…(3) and w = 0... (4). Let A =
1
2
3
4
5
6
7
8
9
10
11
12
0
0
0
1
We will reduce A to its RREF as under:
Add -5 times the 1st row to the 2nd row
Add -9 times the 1st row to the 3rd row
Multiply the 2nd row by -1/4
Add 8 times the 2nd row to the 3rd row
Interchange the 3rd row and the 4th row
Add -3 times the 3rd row to the 2nd row
Add -4 times the 3rd row to the 1st row
Add -2 times the 2nd row to the 1st row
Then the RREF of A is
1
0
-1
0
0
1
2
0
0
0
0
1
0
0
0
0
Thus, the above system of linear equations is equivalent to x –z = 0, y +2z = 0 and w = 0. Hence X = (x,y,z,w) = (z, -2z,z,0) = z(1,-2,1,0). Therefore, W = span{(1,-2,1,0) }. The dimension of W is 1.
| 1 | 2 | 3 | 4 |
| 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 |
| 0 | 0 | 0 | 1 |
![Let W be the subspace of R^4 spanned by [1 2 3 4], [5 6 7 8], [9 10 11 12], [0 0 0 1]. What is the dimension of the orthogonal complement W ? SolutionLet X = (](/WebImages/41/let-w-be-the-subspace-of-r4-spanned-by-1-2-3-4-5-6-7-8-9-10-1128151-1761602046-0.webp "Let W be the subspace of R^4 spanned by [1 2 3 4], [5 6 7 8], [9 10 11 12], [0 0 0 1]. What is the dimension of the orthogonal complement W ? SolutionLet X = (")
![Let W be the subspace of R^4 spanned by [1 2 3 4], [5 6 7 8], [9 10 11 12], [0 0 0 1]. What is the dimension of the orthogonal complement W ? SolutionLet X = (](/WebImages/41/let-w-be-the-subspace-of-r4-spanned-by-1-2-3-4-5-6-7-8-9-10-1128151-1761602046-1.webp "Let W be the subspace of R^4 spanned by [1 2 3 4], [5 6 7 8], [9 10 11 12], [0 0 0 1]. What is the dimension of the orthogonal complement W ? SolutionLet X = (")
