QuestionFind a subset of the following set of vectors that forms a basis for the span(S).
Studdy Solution
STEP 1
1. A basis for the span of a set of vectors is a linearly independent subset of those vectors that spans the same space.
2. To find a basis, we need to check for linear independence among the vectors.
3. We can use row reduction on a matrix formed by these vectors to determine linear independence.
STEP 2
1. Form a matrix with the given vectors as rows.
2. Perform row reduction to echelon form.
3. Identify the pivot rows.
4. Select the corresponding vectors that form a basis.
STEP 3
Form a matrix with the given vectors as rows:
STEP 4
Perform row reduction to echelon form. We will use Gaussian elimination to simplify the matrix:
1. Use the first row to eliminate the first column entries below the pivot (1,0,-2,3).
2. Use the second row to eliminate the second column entries below the pivot (0,1,2,3).
After performing these operations, the matrix becomes:
STEP 5
Continue row reduction to further simplify:
1. Use the third row to eliminate the third column entry below the pivot (-4,-6).
After performing these operations, the matrix becomes:
STEP 6
Identify the pivot rows. The pivot positions are in the first, second, and third rows.
STEP 7
Select the corresponding vectors that form a basis. The vectors corresponding to the pivot rows are:
, , and .
These vectors form a basis for the span of the original set.
The subset of vectors that forms a basis for the span(S) is .
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