MATH 129A
Chapter 2 Summary
Oct. 13, 2015
1. A is an m x n matrix
If B is an ___ x ___matrix then AB is an ___ x ___ matrix.
The columns of AB are __________________________________
ABij

A 

T
ij
( AT ) T 
( AB) T  __________________
Col(A) is ______________________________________________
A basis for Col(A) is _____________________________________
Dim(Col(A))= __________________________________________
Rank(A) is _____________________________________________
Nul(A) is ______________________________________________
A basis for Nul(A) is _____________________________________
Dim(Nul(A))=___________________________________________
2. A is an n x n matrix
A 1 is
A is invertible if and only if
Ax = 0 ________________________________________________
Ax = b ________________________________________________
A has _____ pivots
T(x) = Ax is _____________________________________________
T(x) is _____________________ and T 1 ( x) 
A T is ______________________ and ( AT ) 1 
Col(A) is ____________________ dim(Col(A)) = _______________
Nul(A) is_____________________ dim(Nul(A)) = ______________ Spanning set of H is
If B is also __________________________, then ( AB) 1  ____________
3. Vector spaces
A set V is a vector space if _________________________________
A vector is ______________________________________________
A scalar is _______________________________________________
A subset H of V is a subspace if ______________________________
A span of a set of vectors is _________________________________
A spanning set of H is ______________________________________
A basis of H is ____________________________________________
True/False
Two linearly independent vectors in R n form a 2 dim subspace.
Two vectors in R n form a basis of a 2 dim subspace.
Two linearly independent vectors in R n form a basis of a 2 dim subspace.
Five vectors can never span 2 dim subspace of R n .
Give a few examples of subsets of R n that are not subspaces.
Give a few examples of subsets of R n that are subspaces.