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I am having trouble understanding how to implement the four fundamental subspaces. I have read about the subject and understand the meaning but do not understand how to implement it when asked in a question. Can someone help me better grasp this.

An example question would be:

For each Matrix, accurately sketch the four fundamental subspaces on two R^2 plots so that the orthogonal pairs of subspaces are plotted together. This is the grid picture:

  |1 2|      |1 0|
A=|3 6|    B=|3 0|


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OK, so start by finding the four fundamental subspaces. What are they?

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So Im looking for the column space, null space, row space and left nullspace correct? – user081608 Oct 25 '13 at 5:22
Yes. So, let's start with the column space. What is it for $A$? – Robert Israel Oct 25 '13 at 6:46
Would the new matrix be rows [1 0] and [0 0]? – user081608 Oct 25 '13 at 17:05
So we get C(A) = 1, C(A^T) = 1, Nullspace = x2[01], left Null = 1 – user081608 Oct 25 '13 at 17:19
Column space is by definition the linear span of the columns. In the case of $A$, the second column is $2$ times the first, so the column space is the span of $\pmatrix{1\cr 3\cr}$. – Robert Israel Oct 27 '13 at 2:03

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