Week 4: March 1-5

Chapter 3.2 Subspaces, bases and linear independence

Key points: a) Subspace.

b) Linear dependence and independence; linear relation.

c) Definition of a basis, 3.2.7.

Homework: 18, 24, 32, 38, 46, 50, 39*, 42*.

Chapter 3.3 Dimension

Key Points: a) The maximum number of linearly independent vectors in a

subspace is no greater than the minimum number needed to span the subspace.

b) The notion of dimension.

c) Characterizing a basis of Rⁿ in terms of the invertibility of a

matrix, Fact 3.3.10.

Homework: 22, 30, 32, 38, 44, 53*, 56*

Chapter 3.4 Coordinates

Key points: a) Definition of coordinates with respect to a given basis

b) The matrix of a linear transformation with respect to a given

basis.

c) The relation between the standard matrix of a transformation and

the matrix with respect to some other basis.

d) Remark that a non-standard basis are used make a transformation

look simpler, and this will be a recurring theme as the course procedes.

e) Point out that the ‘standard basis’ is a convention anyway

(consider Australians and the x-y-z basis for 3-space.)

Homework: 4, 14, 18, 32, 36, 29*, 44*