Question

Find a bais for the subspace span by the vectors. What is the dimension of the subspace? $\left(\begin{array}{c}-3 1 0 16\end{array}\right],\left[\begin{array}{c}-2 11 21 3\end{array}\right], \left[\begin{array}{c}0 \\ -6 3\end{array}\right]$ b. Suppose a $4 \times 7$ matrix has 3 pivot columns is $\mathrm{Col} \mathrm{A}=R^{3}$. What the dimension of the Nul A? Explain your answer.CS.SD. 125

Answer 1

To find a basis for the subspace spanned by the given vectors, check if the vectors are linearly independent by forming a matrix and reducing it. The dimension of the subspace is the number of linearly independent vectors in the basis.

Step-by-step explanation:

To find a basis for the subspace spanned by the given vectors, we need to determine if the vectors are linearly independent. We can form a matrix using the given vectors as rows or columns, then row-reduce or column-reduce it to see if it has linearly independent rows or columns. If the matrix has full rank, then the vectors are linearly independent and form a basis for the subspace. If the matrix has a row or column of zeros, then the vectors are linearly dependent and we can remove the zero row or column to find a basis.

The dimension of the subspace is equal to the number of linearly independent vectors in the basis. So, if the vectors are linearly independent and form a basis, the dimension of the subspace is the number of given vectors. If the vectors are linearly dependent and we remove a zero row or column to find a basis, the dimension of the subspace is the number of nonzero vectors in the basis.