Page 123 - 35Linear Algebra

7.1 Linear Transformations and Matrices 123

It is natural to assign these the order: e 1 is first and e 2 is second. An arbitrary vector v
2
of R can be written as

x
v = = xe 1 + ye 2 .
y
To emphasize that we are using the standard basis we define the list (or ordered set)
E = (e 1 , e 2 ) ,

and write

x x
:= (e 1 , e 2 ) := xe 1 + ye 2 = v.
y y
E
You should read this equation by saying:

x
“The column vector of the vector v in the basis E is .”
y
Again, the first notation of a column vector with a subscript E refers to the vector
obtained by multiplying each basis vector by the corresponding scalar listed in the
column and then summing these, i.e. xe 1 +ye 2 . The second notation denotes exactly
the same thing but we first list the basis elements and then the column vector; a
useful trick because this can be read in the same way as matrix multiplication of a row
vector times a column vector–except that the entries of the row vector are themselves
vectors!

n
You should already try to write down the standard basis vectors for R
n
for other values of n and express an arbitrary vector in R in terms of them.
The last example probably seems pedantic because column vectors are al-
ready just ordered lists of numbers and the basis notation has simply allowed
us to “re-express” these as lists of numbers. Of course, this objection does
not apply to more complicated vector spaces like our first matrix example.
Moreover, as we saw earlier, there are infinitely many other pairs of vectors
2
in R that form a basis.
2 {1,2}
Example 76 (A Non-Standard Basis of R = R )

1 1
b = , β = .
1 −1
As functions of {1, 2} they read

1 if k = 1 1 if k = 1
b(k) = , β(k) =
1 if k = 2 −1 if k = 2 .
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