Example: Find the dimension of the subspace of $R^4$ spanned by vectors
$x_1 = (1,2,-1,0)^T$, $x_2 = (2, 5, -3,2)^T$, $x_3 = (2,4,-2,0)^T$, $x_4=(3,8,-5,4)^T$
(T meaning transpose)
Solution: The subspace $Span(x_1,x_2,x_3,x_4)$ is the same as the column space of the matrix $X = [x_1,x_2,x_3,x_4]$
Solution goes on to find row echelon form of $X$ to get the dimension of the column space.
So dimension of $Span(x_1,x_2,x_3,x_4)$ equals $2$.
I am confused because we are asked to find the dimension of the subspace, but the solution finds the dimension of the column space which is the same as finding the rank. How are these two steps related?
The subspace is the span of the $4$ vectors.
We put these $4$ vectors as columns of a matrix and consider its column space. The column space is just the span of the columns of a matrix, but these columns are just those $4$ vectors that we are interested in. Hence the column space of this matrix is equal to the subspace that we are interested in.