Dimensions and Basis of matrices

Say we have a matrix $A_{m\times n}\in\mathbb{R}^{m\times n}$

Column Space

We know that the column space of matrix $A$ lives in $R^{m}$.

But what are the basis for that column space?

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Basis are those (minimum) vectors whose linear combination gives us the desired space.
for example, for space of $R^2$, linear combinations of any $2$ non-parallel(independent) vectors will give us $R^2$

So

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The basis for column space are those (independent)pivot columns

What is the Dimension of column space?

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Number of dimensions are defined by it's basis vectors so, Number of dimensions $=$ number of pivot columns $=$ rank$(r)$ so,
# Dimensions $= r$

Null Space

Null space of matrix $A$ also lives in $R^{m}$.

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Recall those free variables of matrix $A$, free variables are the dependent
column vectors of matrix $A$, these free variables gives us the special solutions
and linear combination of those special solutions gives us our Null space.

What are the basis for that Null space?

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We discussed that Null space is the linear combination of those special solutions so,
The basis for Null space are those special solutions.

What is the Dimension of Null space?

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Number of dimensions are defined by it's basis vectors so, Number of dimensions $=$ number of special solution and we discussed in Null Space section that, Number of special solution are $n-r$.
# Dimensions $= n-r$