Find the basis for the span of the set of vectors \( \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 5 \\ 2 \\ 1 \end{bmatrix} \right\} \).
Answer
Step 3: The non-zero columns of the row echelon form correspond to the basis vectors. Therefore, the basis for the span of the set of vectors is \( \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \\ 0 \\ 0 \end{bmatrix} \right\} \).
Step 1 ：Step 1: Put the vectors as the columns of a matrix: \( A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 5 \\ 1 & 1 & 2 \\ 0 & 1 & 1 \end{bmatrix} \).
Step 2 ：Step 2: Perform Gaussian elimination on the matrix to obtain its row echelon form: \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \).
Step 3 ：Step 3: The non-zero columns of the row echelon form correspond to the basis vectors. Therefore, the basis for the span of the set of vectors is \( \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ -1 \\ 0 \\ 0 \end{bmatrix} \right\} \).
