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Linear Algebra

Clearing the cruff

At this point, we have constructed a number of objects that feel similar:

  • The space $\mathbb{R}^n$.
  • The null space $\mathcal Z(A)$ of a matrix.
  • The range $\mathcal R(A)$ of a matrix.

Geometrically, they all correspond to hyperplanes in $\mathbb{R}^n$ and they all share similar properties.

As an example, we have already seen any range can be expressed as a null space and that any null space can be expressed a range.

Similarly, the set of $t$ free variables found in solving $A \mathbf{x} = \mathbf{0}$ feel very much like vectors in $\mathbb{R}^t$. We can plug in any values for those free variables and get a solution. There is no constraint on them. Much like $\mathbb{R}^t$ is the set of $t$ values in $\mathbb{R}$ with no constraints on them.

In mathematics, when confronted with the many similar-seeming objects that are not all exactly the same in our current framework, it makes sense to search for an abstraction that can unify all the objects as instances of a single idea.

That is what vector spaces are. Almost everything we have seen is an element of a vector space.

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