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Is this a spanning set of R³ (Linear Algebra)?
{(2,1,-2)T, (3,2,-2)T, (2,2,0)T}
the "T" stands for transpose
I did gaussian elimination and worked it down to this (i think its right)
(4x1-2x2-3x3)*(2,1,-2)T +(-2x1-2x2-3x3)*(3,2,-2)T + ((x1+x2)/2)(2,2,0)T
And this is where i get stuck... how can i tell if this is a span of R³? What do i look for? what do i do from here? any help would be great!
1 Answer
- EMLv 71 decade agoFavorite Answer
Another way to solve this is to see if the determinant of the 3x3 matrix of the 3 vectors equals zero. If the determinant equals zero, the vectors are linearly dependent, so they do not span R3.
| 2.. 1.. -2|
| 3.. 2.. -2| =
| 2.. 2... 0|
(2)[(2)(0) - (-2)(2)] - (1)[(3)(0) - (-2)(2)] + (-2)[(3)(2) - (2)(2)] =
8 - 4 - 4 =
0
The vectors do not span R3.
