A First Course in Linear Algebra, 2017a

4.10. Spanning, Linear Independence and Basis in R n 195
Example 4.71: Linear Dependence
Consider the vectors {[
1
4
] [
2
,
3
] [
3
,
2
]}
Are these vectors linearly independent?
Solution. This set contains three vectors in R 2 . By Corollary 4.70 these vectors are linearly dependent. In
fact, we can write
[ ] [ ] [ ]
1 2 3
(−1) +(2) =
4 3 2
showing that this set is linearly dependent.
♠
The third vector in the previous example is in the span of the first two vectors. We could find a way to
write this vector as a linear combination of the other two vectors. It turns out that the linear combination
which we found is the only one, provided that the set is linearly independent.
Theorem 4.72: Unique Linear Combination
Let U ⊆ R n be an independent set. Then any vector⃗x ∈ span(U) can be written uniquely as a linear
combination of vectors of U.
Proof. To prove this theorem, we will show that two linear combinations of vectors in U that equal⃗x must
be the same. Let U = {⃗u 1 ,⃗u 2 ,...,⃗u k }. Suppose that there is a vector⃗x ∈ span(U) such that
⃗x = s 1 ⃗u 1 + s 2 ⃗u 2 + ···+ s k ⃗u k ,forsomes 1 ,s 2 ,...,s k ∈ R, and
⃗x = t 1 ⃗u 1 +t 2 ⃗u 2 + ···+t k ⃗u k ,forsomet 1 ,t 2 ,...,t k ∈ R.
Then⃗0 n =⃗x −⃗x =(s 1 −t 1 )⃗u 1 +(s 2 −t 2 )⃗u 2 + ···+(s k −t k )⃗u k .
Since U is independent, the only linear combination that vanishes is the trivial one, so s i − t i = 0for
all i,1≤ i ≤ k.
Therefore, s i = t i for all i, 1≤ i ≤ k, and the representation is unique. Let U ⊆ R n be an independent
set. Then any vector⃗x ∈ span(U) can be written uniquely as a linear combination of vectors of U. ♠
Suppose that ⃗u,⃗v and ⃗w are nonzero vectors in R 3 ,andthat{⃗v,⃗w} is independent. Consider the set
{⃗u,⃗v,⃗w}. When can we know that this set is independent? It turns out that this follows exactly when
⃗u ∉ span{⃗v,⃗w}.
Example 4.73:
Suppose that⃗u,⃗v and ⃗w are nonzero vectors in R 3 ,andthat{⃗v,⃗w} is independent. Prove that {⃗u,⃗v,⃗w}
is independent if and only if ⃗u ∉ span{⃗v,⃗w}.
Solution. If⃗u ∈ span{⃗v,⃗w}, then there exist a,b ∈ R so that⃗u = a⃗v+b⃗w. This implies that⃗u−a⃗v−b⃗w =⃗0 3 ,
so ⃗u−a⃗v − b⃗w is a nontrivial linear combination of {⃗u,⃗v,⃗w} that vanishes, and thus {⃗u,⃗v,⃗w} is dependent.