Let $A$ be the following $4 \times 4$ matrix. \begin{bmatrix}2&3&2&0\\1&1&1&-1\\3&3&3&-5\\2&3&2&0\end{bmatrix}

Without mentioning pivots (columns or entries) explain why col(A) $\neq$ R4

I know the answer to this question but I am not sure why? Can someone please explain this, so I can understand why col(A) is not R4?

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    $\begingroup$ It can be seen that two columns are equal here. If I am not allowed to refer to the rows and columns here as you are saying, then what can I possibly refer to? $\endgroup$ May 2 '17 at 0:16

You can generate the whole space $\mathbb{R}^4$ only when you have four Linearly Independent vectors from $\mathbb{R}^4$. There are four column vectors from the matrix, that's very fine. But the bad thing about them is that they are not Linearly Independent, because column $1$ is equal to column $2$. So they can't generate the $\mathbb{R}^4$


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