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6.4 The Vector Space R n 173

form another basis B 2 for S. Why does B 1 seem more natural than B 2 ? It is because, given any X
in S, it is easy to find the coordinates of X with respect to B 1 . Indeed, if X =< a,b,0,0 >, then
immediately
X = ae 1 + be 2 .

However, finding the coordinates of X with respect to B 2 is more tedious. If these coordinates are
c 1 and c 2 , then we would have to have

< a,b,0,0 > = c 1 w 1 + c 2 w 2
= c 1 < 2,−6,0,0 > +c 2 < 2,4,0,0 >
=< 2c 1 + 2c 2 ,−6c 1 + 4c 2 ,0,0 >,

requiring that
2c 1 + 2c 2 = a and − 6c 1 + 4c 2 = b.
Solve for these coordinates to obtain
1 1
c 1 = (2a − b),c 2 = (3a + b).
10 10
Thus,
1 1
X = (2a − b)w 1 + (3a + b)w 2 .
10 10
We can tell the coordinates of any X in S with respect to B 1 just by looking at X, while finding
the coordinates of X with respect to B 2 takes some work.
Another nice feature of B 1 is that it consists of mutually orthogonal vectors. In general, a
basis is an orthogonal basis if its vectors are mutually orthogonal. If these vectors are also unit
vectors, then the basis is orthonormal. With any orthogonal basis for S, it is possible to write a
simple formula for the coordinates of any vector X in S.

THEOREM 6.5 Coordinates in Orthogonal Bases

n
Let S be a subspace of R and let V 1 ,··· ,V k be an orthogonal basis for S.If X is in S, then
X = c 1 V 1 + c 2 v 2 + ··· + c k V k ,
where
X · V j
c j =
V j 2
for j = 1,2,··· ,k.
This gives the jth coordinate of any X with respect to these basis vectors as the dot product
of X with V j , divided by the length of V j squared. In terms of projections, any vector in X is
the sum of the projections of X onto the orthogonal basis vectors. This is true for any orthogonal
basis for S.

Proof Write
X = c 1 V 1 + c 2 V 2 + ··· + c k V k .
We must solve for the c j s. Take the dot product of X with V j to obtain

2
X · V j = c j V j · V j = c j V j ,
since, by orthogonality, V i · V j = 0if i = j. This yields the expression for c j given in the
theorem.

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October 14, 2010 14:21 THM/NEIL Page-173 27410_06_ch06_p145-186