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title: "Axler 3.B"
source: https://www.jemoka.com/posts/kbhaxler_3_b/
---

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 homogenous system with more variables than equations has nonzero solutions inhomogenous system with more equations than variables has no solutions for an arbitrary set of constants New Definitions null space injectivity range surjectivity homogeneous system 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 &ldquo;sizes&rdquo; of maps map to smaller space is not injective map to bigger space is not surjective solving systems of equations: homogenous system with more variables than equations has nonzero solutions inhomogenous system with more equations than variables has no solutions for an arbitrary set of constants Questions for Jana &ldquo;To prove the inclusion in the other direction, suppose v 2 null T.&rdquo; for 3.16; what is the first direction? maybe nothing maps to \(0\)
