## Key Sequence

- we defined the null space and injectivity
- from that, we showed that injectivity IFF implies that null space is \(\{0\}\), essentially because if \(T0=0\) already, there cannot be another one that also is taken to \(0\) in an injective function

- we defined range and surjectivity
- we showed that these concepts are strongly related by the fundamental theorem of linear maps: if \(T \in \mathcal{L}(V,W)\), then \(\dim V = \dim null\ T + \dim range\ T\)
- from the fundamental theorem, we showed the somewhat intuitive pair about the sizes of maps: map to smaller space is not injective, map to bigger space is not surjective
- we then applied that result to show results about homogeneous systems

## New Definitions

## Results and Their Proofs

- the null space is a subspace of the domain
- injectivity IFF implies that null space is \(\{0\}\)
- the fundamental theorem of linear maps
- “sizes” of maps
- solving systems of equations:

## Questions for Jana

~~“To prove the inclusion in the other direction, suppose v 2 null T.” for 3.16; what is the~~maybe nothing maps to \(0\)*first*direction?