finding columnspace for a matrix

Question

$$\begin{pmatrix}1 & 0 & 0 & 1 & -1 \\ 0 & 1 & 0 & -1 & -2 \\ 0 & 0 & 1 & 0 & 0\end{pmatrix}$$

For the matrix above, can we find the column space even we do not have the original matrix before performing Gaussian elimination?

Welcome to Mathematics Stack Exchange. The column space is in $\mathbb R^3$, and it is all of $\mathbb R^3$ because the first three columns of that matrix are the standard basis for $\mathbb R^3$ and therefore span $\mathbb R^3$ — J. W. Tanner, Mar 13 '20 at 03:34
Can I know when I need to refer back to the original matrix and when I don't need? — newbie, Mar 13 '20 at 03:38
Perhaps I misunderstood your question. Gaussian elimination could change the column space — J. W. Tanner, Mar 13 '20 at 03:42

score 0 · Accepted Answer · answered Mar 13 '20 at 03:53

0

For your particular matrix, the rank is $3$. Since the column length is $3$, the column space must be $\mathbb{R}^3$.

In general, we do need the original matrix but this is a special case.

answered Mar 13 '20 at 03:53

Siong Thye Goh

So can I say that when the matrix is full rank, I can directly find the column space without referring back to the original matrix? – newbie Mar 13 '20 at 04:07
@newbie You can if the matrix is at least as wide as it is tall. – amd Mar 13 '20 at 04:26

score 0 · Answer 2 · answered Mar 13 '20 at 06:32

0

Yes! Look at the first three columns. They are the standard basis vectors for $\mathbb{R}^3$.

answered Mar 13 '20 at 06:32

Julian L

2 Answers2