Column space

Column space

Wiki: In linear algebra, the column space of a matrix A=[\(a_1,a_2,\cdots,a_n\)] is the span (all possible linear combinations) of its column vectors and is denoted as \(C(A)\)=span({\(a_1,a_2,\cdots,a_n\)}). The row space is similar which is denoted as \(C(A^T)\).

Any linear combination of column vectors of matrix A can be written the product of A with a column vector:

\[Ax = A\begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix}=\begin{bmatrix}a_{11}&\cdots&a_{1n}\\\vdots&\ddots&\vdots\\a_{m1}&\cdots&a_{mn}\end{bmatrix}\begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix}\\=\begin{bmatrix}x_1a_{11}+\cdots+x_na_{1n}\\\vdots\\x_1a_{m1}+\cdots+x_na_{mn}\end{bmatrix}=x_1\begin{bmatrix}a_{11}\\\vdots\\a_{m1}\end{bmatrix}+\cdots+x_n\begin{bmatrix}a_{1n}\\\vdots\\a_{mn}\end{bmatrix}\\ =x_1a_1+\cdots+x_na_n \]

So the x here indicates all the possible linear combinations of column vectors of matrix A.

And for the linear equation \(Ax=b\), if it establishes, it means that vector b is actually in within column space spanned by column vectors of A (\(b\in \text{colsp(A)}\)) because we can find a solution ( a set of linear combination coefficient that makes b equals to Ax.)