Notes  Linear Algebra I MT22, Chapter 4

Content covered:
 Proof coefficients of a linear combination of basis vectors are unique (Prop. 99)
 Proof a linearly independent set of vectors can’t contain duplicates (Lem. 102)
 Prove that a necessary and sufficient condition for a set to be a basis is that every vector has a unique expression as a linear combination of its elements (Prop. 111).
 Prove lots of properties about linear independence in RREF (Prop. 117, Cor. 118, Cor. 119)
 Prove that RREF is unique (The. 122)
 Prove a $Ax = b$ is consistent if and only if $\text{rank}(A \vert b) = \text{rank}(A)$ (Cor 124.)
 Prove lots of properties about the number of solutions of a system of equations and the rank of the matrix representing it.