Page 117 - 35Linear Algebra
P. 117
6.4 Bases (Take 1) 117
sum of multiples and then applying linearity;
x x + y 1 x − y 1
L = L +
y 2 1 2 −1
x + y 1 x − y 1
= L + L
2 1 2 −1
x + y 2 x − y 6
= +
2 4 2 8
x + y 3(x − y)
= +
2(x + y) 4(x − y)
4x − 2y
=
6x − y
Thus L is completely specified by its value at just two inputs.
It should not surprise you to learn there are infinitely many pairs of
2
vectors from R with the property that any vector can be expressed as a
linear combination of them; any pair that when used as columns of a matrix
2
gives an invertible matrix works. Such a pair is called a basis for R .
Similarly, there are infinitely many triples of vectors with the property
3
that any vector from R can be expressed as a linear combination of them:
these are the triples that used as columns of a matrix give an invertible
3
matrix. Such a triple is called a basis for R .
In a similar spirit, there are infinitely many pairs of vectors with the
property that every vector in
1 0
1
V = c 1 + c 2 c 1 , c 2 ∈ R
1
0 1
can be expressed as a linear combination of them. Some examples are
1 0 1 1
3
1
V = c 1 + c 2 c 1 , c 2 ∈ R = c 1 + c 2 c 1 , c 2 ∈ R
1
2
0 2 0 2
Such a pair is a called a basis for V .
You probably have some intuitive notion of what dimension means (the
careful mathematical definition is given in chapter 11). Roughly speaking,
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