Page 217 - 35Linear Algebra

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11.1 Bases in R . 217

n
n
a basis for R , and dim R = n. This basis is often called the standard
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or canonical basis for R . The vector with a one in the ith position and
zeros everywhere else is written e i . (You could also view it as the function
{1, 2, . . . , n} → R where e i (j) = 1 if i = j and 0 if i 6= j.) It points in the
direction of the ith coordinate axis, and has unit length. In multivariable
ˆ ˆ ˆ
3
calculus classes, this basis is often written {i, j, k} for R .
Note that it is often convenient to order basis elements, so rather than
writing a set of vectors, we would write a list. This is called an ordered
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basis. For example, the canonical ordered basis for R is (e 1 , e 2 , . . . , e n ). The
possibility to reorder basis vectors is not the only way in which bases are
non-unique.

Bases are not unique. While there exists a unique way to express a vector in terms
of any particular basis, bases themselves are far from unique. For example, both of
the sets

1 0 1 1
, and ,
0 1 1 −1
2
are bases for R . Rescaling any vector in one of these sets is already enough to show
2
that R has infinitely many bases. But even if we require that all of the basis vectors
2
have unit length, it turns out that there are still infinitely many bases for R (see
review question 3).

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To see whether a set of vectors S = {v 1 , . . . , v m } is a basis for R , we have
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to check that the elements are linearly independent and that they span R .
From the previous discussion, we also know that m must equal n, so lets
assume S has n vectors. If S is linearly independent, then there is no non-
trivial solution of the equation

1
n
0 = x v 1 + · · · + x v n .
Let M be a matrix whose columns are the vectors v i and X the column
i
vector with entries x . Then the above equation is equivalent to requiring
that there is a unique solution to

MX = 0 .

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To see if S spans R , we take an arbitrary vector w and solve the linear
system
1 n
w = x v 1 + · · · + x v n

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