Linear Algebra

Section II. Linear Independence 109
Proof. If the finite set S is linearly independent then there is nothing to prove
so assume that S = {�s1,...,�sn} is linearly dependent. By Corollary 1.16, there
is a vector �si that is a linear combination of �s1, ... , �si−1. Define S1 to be the
set S −{�si}. Lemma 1.1 then says that the span does not shrink: [S1] =[S].
If S1 is linearly independent then we are finished. Otherwise repeat the
prior paragraph to derive S2 ⊂ S1 such that [S2] =[S1]. Repeat this process
until a linearly independent set appears; one must eventually appear because
S is finite. (Formally, this part of the argument uses mathematical induction.
Exercise 37 asks for the details.) QED
In summary, we have introduced the definition of linear independence to
formalize the idea of the minimality of a spanning set. We have developed some
elementary properties of this idea. The most important is Lemma 1.15, which,
complementing that a spanning set is minimal when it linearly independent,
tells us that a linearly independent set is maximal when it spans the space.
Exercises
� 1.18 Decide whether each subset of R 3 is linearly dependent or linearly independent.
� �
1
� �
2
� �
4
(a) { −3 , 2 , −4 }
5
� �
1
4
� �
2
14
� �
3
(b) { 7 , 7 , 7 }
7
� �
0
7
� �
1
7
(c) { 0 , 0 }
−1
� �
9
4
� �
2
� �
3
� �
12
(d) { 9 , 0 , 5 , 12 }
0 1 −4 −1
� 1.19 Which of these subsets of P3 are linearly dependent and which are independent?
(a) {3 − x +9x 2 , 5 − 6x +3x 2 , 1+1x− 5x 2 }
(b) {−x 2 , 1+4x 2 }
(c) {2+x +7x 2 , 3 − x +2x 2 , 4 − 3x 2 }
(d) {8+3x +3x 2 ,x+2x 2 , 2+2x +2x 2 , 8 − 2x +5x 2 }
� 1.20 Prove that each set {f,g} is linearly independent in the vector space of all
functions from R + to R.
(a) f(x) =x and g(x) =1/x
(b) f(x) = cos(x) andg(x) =sin(x)
(c) f(x) =e x and g(x) =ln(x)
� 1.21 Which of these subsets of the space of real-valued functions of one real variable
is linearly dependent and which is linearly independent? (Note that we have
abbreviated some constant functions; e.g., in the first item, the ‘2’ stands for the
constant function f(x) =2.)