Fundamental Theorem of Linear Maps!

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1. Intuition
2. #definition Fundamental Theorem of Linear Maps def
- 2.1. See also dimension, null space, and range of a function.
3. AKA: Rank Nullity Theorem

1 Intuition

In a linear map $T : V\to W$, the dimension of the domain $V$ is amount of stuff that you throw away (null space) plus the amount of stuff that does not get thrown away (the column space). If $T$ is a map from $V$ to $W$, then the dimension of the source map $V$ is the dimension of the null space (everything in $V$ that $T$ takes to 0) plus the dimension of the range (all possible things taken to by $T$)

2 #definition Fundamental Theorem of Linear Maps def

Suppose $V$ is finite-dimensional and $T ∈ \mathcal L(V, W). Then \text{range }T is finite-dimensional and \[ \text{dim }V = \text{dim null }T + \text{dim range} T \]

2.1 See also dimension, null space, and range of a function.

3 AKA: Rank Nullity Theorem

This is a subset of the Fundamental Theorem of Linear Algebra (#todo-expand)