LINEAR INDEPENDENCE [1.7]
Linear independence
Definition
äThe set {v1, ..., vp}is said to be linearly dependent if
there exist weights c1, ..., cp, not all zero, such that
c1v1+c2v2+... +cpvp= 0
äIt is linearly independent otherwise
äThe above equation is called linear dependence relation
among the vectors v1,··· , vp
äAnother way to express dependence: A set of vectors is
linearly dependent if and only if there is one vector among
them which is a linear combination of all the others.
-Prove this
F-2
Q: Why do we care about linear independence?
A: When expressing a vector xas a linear combination of
a system {v1,··· , vp}that is linearly dependent, then we
can find a smaller system in which we can express x
äA dependent system is ‘redundant’
-Let v1="1
1#. Is {v1}linearly independent? [special
case where p= 1
-A system consisting of a nonzero vector [at least one
nonzero entry] is always linearly independent: True - False?
-Are the following systems linearly independent:
("1
0#,"−1
1#),("1
0#,"−10
0#),("1
0#,"−1
1#,"2
1#)?
F-3
-Let v1=
1
1
2
;v2=
4
1
5
;v3=
−2
3
1
;
(a) Determine if {v1, v2, v3}is linearly independent
(b) If possible find a linear dependence relation among
v1, v2, v3.
Solution: We must determine if the system:
x1
1
1
2
+x2
4
1
5
+x3
−2
3
1
=
0
0
0
has a nontrivial solution
Note Solution is trivial when x1=x2=x3= 0
F-4
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