MATH 348 - Advanced Engineering Mathematics April 21, 2008
Homework 9 - Hints
Definition - Linear Combination - Given a set, S, of nvectors,
S={a1,a2,a3,...,an}.(1)
We say that bis a linear combination of the vectors from Sif there exist scalars x1, x2, x3, . . . , xnsuch that,
b=
n
X
i=1
xiai.(2)
Definition - Linear Independence - Given a set, S, of nvectors,
S={v1,v2,v3,...,vn}.(3)
We say that the vectors of Sforms a linearly independent set if and only if the scalars x1=x2=x3=· · · =xn= 0
are the only solution to,
0=
n
X
i=1
xivi.(4)
If a set of vectors does not form a linearly independent set then the vectors are said to be linearly dependent.
Definition - Column Space - Given the matrix A∈Rm×nwe define the set of all linear combinations of the columns
of the matrix Ato be the column space of Aand we denote this space of vectors as Col(A).
Definition - Null Space - Given the matrix A∈Rm×nwe define the set of vectors x, which satisfies the equation
Ax =0to be the Null Space of Aand we denote this space of vectors Nul(A).
Note - Straightforward calculations will verify that,
n
X
i=1
xiai=x1
a11
a21
a31
.
.
.
am1
+x2
a12
a22
a32
.
.
.
am2
+x3
a13
a23
a33
.
.
.
am3
+· · · +xn
a1n
a2n
a3n
.
.
.
amn
=Ax (5)
Hints:
•Problem 3 - If you are given Ax =band a solution xexists then what does this imply about bin light of (2)?
•Problem 4 - If x=0is the only solution to Vx =0then (5) implies that the columns of Vforms a linearly
independent set.
•Problem 5 - What is Aw? What does this imply about w? Is there a solution to Ax=w? What does this imply
about w?
1