Key facts:•The Invertible Matrix Theorem (Theorem 8 on page 129). This brings together mostof the ideas we’ve seen so far. The idea is not to memorise this theorem (althoughI suppose this wouldn’t hurt, if you’re the kind of person who likes to memorisethings), but more importantly you should be able to go through it and understandwhy all of the statements are equivalent. The key idea is that ifAhas an inverse,thenAcan be “cancelled” out of equations by multiplying by the inverse.•A linear transformationT:Rn→Rnis invertible if and only if its standard matrixAis invertible. In this case, the standard matrix ofT-1isA-1.Practice:Things you need to be able to do/answer: This section is all about theory.A typicalquestion would be, “Which of the following statements is true/false?” followed by a listof statements about invertible matrices, linear transformations, pivot positions, etc. Toanswer such a question, read through the statements one by one, and see which one(s)must be true, based on the invertible matrix theorem (and other theorems we’ve seen).There’s really not much more to it than that.2.8 Subspaces ofRnTheory:Key terms and ideas:•Subspace ofRn- definition.•Column space and null space of anm×nmatrix.•Basis for a subspace.Key facts:•Any set of the form span{S}is a subspace (Sis any set of vectors).•Col(A)and Nul(A)are subspaces.•The pivot columns ofAform a basis for Col(A).•The vectors appearing in the PVF of the solution set ofAx=0form a basis forNul(A).

Practice:Things you need to be able to do/answer:•“Which of the following sets of vectors is a subspace?” To answer this, check whichone satisfies all three defining properties of a subspace.•“Find a basis for Col(A).” Row-reduceA, and see which are the pivot columns;then goback to Aand take those columns fromAas your basis.•“Find a basis for Nul(A).” Add a column of zeros toA, row-reduce, and write thegeneral solution in PVF. The vectors in this solution are the basis you want.•“Find a basis for the span of the following vectors.”Arrange the vectors as thecolumns of a matrix, then find a basis for the column space of that matrix.2.9 Dimension and rankTheory:Key terms and ideas:•Coordinates of a vector relative to a basis.•Dimension of a subspace.•Rank of a matrix.

Practice:Things you need to be able to do/answer: