Given the following vectors for the matrix A: \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \), what is the column space of A?
Answer
This means that the column space of A is the set of all vectors that can be written in this form.
Step 1 ：The column space of a matrix are all possible linear combinations of its column vectors. So for matrix A, we need to look at each of the column vectors.
Step 2 ：The column vectors of A are \( \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} \), \( \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} \), and \( \begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix} \)
Step 3 ：We can write any vector in the column space as a combination of these vectors, \( x\begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix} + y\begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} + z\begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix} \), where x, y, and z are scalars.
Step 4 ：This means that the column space of A is the set of all vectors that can be written in this form.
