Axler 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
• “sizes” 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

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