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Find a basis for the row space and the rank of the matrix -3 -12 12 5 2 8 -8 -9 -3 -12 12 4?

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Final answer:

To find a basis for the row space and the rank of a matrix, perform row reduction to obtain the row-echelon form. The row space is the span of the non-zero rows in the row-echelon form, and the rank is the number of linearly independent rows.

Step-by-step explanation:

To find a basis for the row space and the rank of a matrix, we need to perform row reduction to obtain the row-echelon form or the reduced row-echelon form of the matrix. Let's perform row reduction on the given matrix:

-3 -12 12

5 2 8

-8 -9 -3

-12 12 4

By performing row reduction, we obtain the following row-echelon form:

1 4 -4

0 3 3

0 0 0

0 0 0

The row space is the span of the non-zero rows in the row-echelon form. In this case, the row space is spanned by the vectors [1, 4, -4] and [0, 3, 3]. Therefore, a basis for the row space is { [1, 4, -4], [0, 3, 3] }.

The rank of a matrix is the number of linearly independent rows or columns. In this case, the rank is equal to the number of non-zero rows in the row-echelon form, which is 2. Therefore, the rank of the given matrix is 2.

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