A First Course in Linear Algebra, 2017a

4.10. Spanning, Linear Independence and Basis in R n 191
Definition 4.64: Linearly Independent Set of Vectors
A set of non-zero vectors {⃗u 1 ,···,⃗u k } in R n is said to be linearly independent if whenever
it follows that each a i = 0.
k
∑
i=1
a i ⃗u i =⃗0
Note also that we require all vectors to be non-zero to form a linearly independent set.
To view this in a more familiar setting, form the n × k matrix A having these vectors as columns. Then
all we are saying is that the set {⃗u 1 ,···,⃗u k } is linearly independent precisely when AX = 0 has only the
trivial solution.
Here is an example.
Example 4.65: Linearly Independent Vectors
Consider the vectors ⃗u = [ 1 1 0 ] T , ⃗v =
[
1 0 1
] T ,and⃗w =
[
0 1 1
] T in R 3 . Verify
whether the set {⃗u,⃗v,⃗w} is linearly independent.
Solution. So suppose that we have a linear combinations a⃗u+b⃗v +c⃗w =⃗0. Then you can see that this can
only happen with a = b = c = 0.
⎡ ⎤
1 1 0
As mentioned above, you can equivalently form the 3 × 3matrixA = ⎣ 1 0 1 ⎦, andshowthat
0 1 1
AX = 0 has only the trivial solution.
Thus this means the set {⃗u,⃗v,⃗w} is linearly independent.
♠
In terms of spanning, a set of vectors is linearly independent if it does not contain unnecessary vectors.
That is, it does not contain a vector which is in the span of the others.
Thus we put all this together in the following important theorem.