Page 124 - 35Linear Algebra

124 Matrices

Notice something important: there is no reason to say that β comes before b or
vice versa. That is, there is no a priori reason to give these basis elements one order
or the other. However, it will be necessary to give the basis elements an order if we
want to use them to encode other vectors. We choose one arbitrarily; let

B = (b, β)

be the ordered basis. Note that for an unordered set we use the {} parentheses while
for lists or ordered sets we use ().
As before we define

x x
:= (b, β) := xb + yβ .
y y
B
You might think that the numbers x and y denote exactly the same vector as in the
previous example. However, they do not. Inserting the actual vectors that b and β
represent we have

1 1 x + y
xb + yβ = x + y = .
1 −1 x − y
Thus, to contrast, we have

x x + y x x
= and =
y x − y y y
B E
Only in the standard basis E does the column vector of v agree with the column vector
that v actually is!

Based on the above example, you might think that our aim would be to
find the “standard basis” for any problem. In fact, this is far from the truth.
Notice, for example that the vector

1
v = = e 1 + e 2 = b
1
written in the standard basis E is just

1
v = ,
1
E
which was easy to calculate. But in the basis B we find

1
v = ,
0
B
124