MATH-332: Linear Algebra 1-1
MATH-332: Linear Algebra Chapter: 07
Linear Equations in Linear Algebra
Section 1.7: Linear Independence
pgs. 65-72 June 22, 2009
Lecture: Linear Independence
Topics:
Linear Combinations
Linear Independence
Characterizations of Linearly Dependent Sets
Problems Prac: 1 - 4
Prob: 9, 15, 17, 19, 21, 27
This is one of the most important sections in the text. Many other linear algebra textbooks leave
this material for much later when it hinders more than it helps. The author can do this by initially
highlighting the four-fold description of a linear system. 1Particularly, the vector description found
in section 1.3 pushes the concept of a linear combination to the forefront and this concept can now
be used to bread the notion of linear independence.
The idea of linear independence of vectors is one of the most fundamental concepts in mathe-
matics. A great example is found in differential equations. Given,
dY
dt =AY,(1)
has a solution that can be described in terms of the linear combination,
Y(t) = k1v1eλ1t+k2v2eλ2t,(2)
so long as the eigenvectors v1,v2are linearly independent. 2This theme is seen in more general
settings where the prescription is this:
1. Determine what ‘vector-space’ you are working with.3
2. Determine a linearly independent set of vectors, which spans the vector space. 4
3. Write down arbitrary vectors from this space as linear combinations of the linearly independent
vectors.
1Recall that this is:
1. The linear system itself.
2. Ax =b
3. [A|b]
4. Pn
i=1 xiai=b
2In a more geometric setting we can say that the differential equation defines a two-dimensional solution space
(this was what we called phase space) and that needs two linearly independent vectors/solutions to span it. Arbitrary
solutions can then be constructed using linear combinations of these ‘basis’ vectors.
3A vector-space is exactly what the name implies, a space full of vectors. The term space implies that we have
a collection of vectors together with some sort of algebra, while the term vector can mean quite a lot of things. See
chapter 4 if you just can’t wait.
4Linearly independent vectors are nice but orthogonal vectors are even better. One can quickly see that the standard
basis vectors of R2are linearly independent and thus any vector from the plane can take the form x=c1ˆ
i+c2ˆ
j, but
since ˆ
i·ˆ
j= 0 things, we will find, are even better.
