QuestionDo the vectors , , and span $\mathbb{R}^{3$? Explain.
Studdy Solution
STEP 1
Assumptions1. The vectors are and
. We need to determine if these vectors span
3. Spanning means that any vector in can be written as a linear combination of these vectors.
STEP 2
We will first write these vectors as the columns of a matrix .
STEP 3
To determine if the vectors span , we need to find the rank of the matrix . The rank of a matrix is the maximum number of linearly independent rows or columns. If the rank of the matrix is3, then the vectors span .
STEP 4
We will perform Gaussian elimination to find the row echelon form of the matrix. We start by swapping the first and third rows to bring a non-zero element to the top.
STEP 5
Next, we multiply the first row by to make the leading element of the first row1.
STEP 6
Then, we multiply the second row by to make the leading element of the second row1.
STEP 7
Finally, we multiply the third row by to make the leading element of the third row1.
STEP 8
Now that we have the row echelon form of the matrix, we can see that the rank of the matrix is3, since there are3 non-zero rows. This means that the vectors span .
So, the correct answer is D. Yes. When the given vectors are written as the columns of a matrix has a pivot position in every row.
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