Questions: Find a basis for the column space and the rank of the matrix. [ 2 4 1 6 ]

Solution Steps

To find a basis for the column space of a matrix, we can perform Gaussian elimination to bring the matrix to its row-echelon form. The pivot columns in this form correspond to the basis vectors for the column space. The rank of the matrix is the number of pivot columns.

We start with the matrix \( A = \begin{bmatrix} 2 & 4 \\ 1 & 6 \end{bmatrix} \).

After performing Gaussian elimination, we find the row-echelon form of the matrix. The pivot columns are identified as \( [0, 1, 1] \).

The basis for the column space is formed by the original columns of the matrix corresponding to the pivot columns. Thus, the basis is given by: \[ \text{Basis} = \begin{bmatrix} 2 & 4 \\ 1 & 6 \end{bmatrix} \]

The rank of the matrix is determined by the number of pivot columns, which is \( 2 \).

Final Answer