From what I've seen online, it seems that the null space isn't in the column space, but I don't understand why that is the case. If the null space is the set of all combinations that equal 0, and the column space is the set of all combinations, would the null space always be inside the column space? I'm talking about a matrix if that isn't clear.
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The null space and the column space aren't in the same space! If $T:V\to W$ then the null space of $T$ is the vectors of $V$ such that the corresponding linear combination of columns of $T$ is $0$, while the column space is the space of linear combinations of columns of $T$, which all lie in $W$.
Kevin Carlson
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The null space isn't the set of combinations of columns that equal zero, it's the set of indices that make that happen. For example, $$\left(\begin{smallmatrix}1&2&0\\0&0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}a\\b\\c\end{smallmatrix}\right)=\left(\begin{smallmatrix}0\\0\end{smallmatrix}\right)$$
The columnspace is the subspace of $\mathbb{R}^2$ generated by $\left\{\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right),\left(\begin{smallmatrix}2\\0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\1\end{smallmatrix}\right)\right\})$, which happens to be all of $\mathbb{R}^2$. The nullspace is a subspace of $\mathbb{R}^3$, namely $$\{(a,b,c):a+2b=0, c=0\}$$ This is a one-dimensional space, with basis $\left\{\left(\begin{smallmatrix}-2\\1\\0\end{smallmatrix}\right)\right\}$.
vadim123
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