A First Course in Linear Algebra, 2017a

192 R n
Theorem 4.66: Linear Independence as a Linear Combination
Let {⃗u 1 ,···,⃗u k } be a collection of vectors in R n . Then the following are equivalent:
1. It is linearly independent, that is whenever
it follows that each coefficient a i = 0.
2. No vector is in the span of the others.
k
∑
i=1
a i ⃗u i =⃗0
3. The system of linear equations AX = 0 has only the trivial solution, where A is the n × k
matrix having these vectors as columns.
The last sentence of this theorem is useful as it allows us to use the reduced row-echelon form of a
matrix to determine if a set of vectors is linearly independent. Let the vectors be columns of a matrix A.
Find the reduced row-echelon form of A. If each column has a leading one, then it follows that the vectors
are linearly independent.
Sometimes we refer to the condition regarding sums as follows: The set of vectors, {⃗u 1 ,···,⃗u k } is
linearly independent if and only if there is no nontrivial linear combination which equals the zero vector.
A nontrivial linear combination is one in which not all the scalars equal zero. Similarly, a trivial linear
combination is one in which all scalars equal zero.
Here is a detailed example in R 4 .
Example 4.67: Linear Independence
Determine whether the set of vectors given by
⎧⎡
⎤ ⎡ ⎤ ⎡
1 2
⎪⎨
⎢ 2
⎥
⎣ 3 ⎦
⎪⎩
, ⎢ 1
⎥
⎣ 0 ⎦ , ⎢
⎣
0 1
0
1
1
2
⎤
⎡
⎥
⎦ , ⎢
is linearly independent. If it is linearly dependent, express one of the vectors as a linear combination
of the others.
⎣
3
2
2
0
⎤⎫
⎪⎬
⎥
⎦
⎪⎭
Solution. In this case the matrix of the corresponding homogeneous system of linear equations is
⎡
⎤
1 2 0 3 0
⎢ 2 1 1 2 0
⎥
⎣ 3 0 1 2 0 ⎦
0 1 2 0 0