Study Guide for Week 3
On Monday we will spend one more lecture on linear
independence (section 1.7). Throughout
the rest of the week we will focus on linear transformations (sections 1.8-1.9)
and basic operations
on matrices (section 2.1). The important topics concerning linear
transformations and matrix
ope rations that will be covered throughout week 3 (Oct. 13-Oct 17) are listed
be low .
• Definition of a linear transformation; in particular
– a transformation T is a vector- valued function that as signs each vector u in
to a
vector T(u) in 
.
– A linear transformation is a transformation that satisfies the additive
property (i)
T(u + v) = T(u) + T(v) and the multiplicative property (ii) T(cu) = cT (u) where
c is
a real number .
You should be able to verify whether a transformation is linear or not. (lecture
9 and section
1.8)
• Any linear transformation T that maps vectors in
to vectors in
can be written in the
form T(x) = Ax where x is in and A is an m ×
n matrix. The matrix A associated with
the transformation T can be constructed by
.
Here ej is the vector in with all
components equal to zero except that its jth component
is one. Notice that these are vectors parallel to the standard coordinate axes .
It is important
to be able to construct the matrix A for a transformation T. (lecture 9 and
section 1.9)
• Conversely any transformation in the form T(x) = Ax is linear. In other words
such a
transformation satisfies additive and multiplicative properties. (lecture 9 and
section 1.8)
• Operations on matrices (exclude the matrix inverse, which will be covered next
week); particularly
you should be able to perform
– matrix-matrix addition, matrix transpose, matrix- matrix product
and know for which matrices these operations are defined and for which they are
not. (lecture
10 and section 2.1)
• Basic properties of the matrix operations above; for instance
– Addition commutes : A + B = B + A.
– Multiplication does not commute : AB ≠ BA in general.
– Multiplication is associative : (AB)C = A(BC)
– Transposing twice gives back the original matrix : (AT )T
= A
– Multiplication distributes over addition : C(A + B) = CA + CB
(lecture 10 and section 2.1)
Homework 3 (due on October 24th, Friday by 3pm)
1. Justify your answers for the questions below.
(a) Can three vectors v1, v2, v3 in
span
?
(b) Can four vectors w1,w2,w3,w4 in
be linearly independent?
(c) Given a 4 × 3 matrix A. Can the system Ax = b be consistent for all b in
?
(d) Given a 3 × 4 matrix A. Can the system Ax = 0 be inconsistent? Can this
system
have a unique solution ?
2. Given the transformations 
and
defined as
(a) Are the transformations T1, T2 and T3
linear or not? Justify your answers.
(b) Ex press T 2 in the form T2(x) = Ax where A is a 3 × 2
matrix.
3. A linear transformation T from 
to
acts on the vectors
and
as follows
Find the transformation matrix A such that T(x) = Ax for
each x in .
4. Consider the matrices
(a) Perform each of the operations AC+B, ACB, BC and BCT
if the operation is defined.
If the operation is not defined, explain why not.
(b) Do the equalities (i) (AC + B) = (B + AC) and (ii) ACB = BAC hold?
