Linear Algebra

258 Chapter 3. Maps Between Spaces
For the i = 2 case, expand the definition of �κ2.
�κ2 = � β2 − proj [�κ1]( � β2) = � β2 − � β2 �κ1
· �κ1 =
�κ1 �κ1
� β2 − � β2 �κ1
·
�κ1 �κ1
� β1
This expansion shows that �κ2 is nonzero or else this would be a non-trivial linear
dependence among the � β’s (it is nontrivial because the coefficient of � β2 is 1) and
also shows that �κ2 is in the desired span. Finally, �κ2 is orthogonal to the only
preceding vector
�κ1 �κ2 = �κ1 ( � β2 − proj [�κ1]( � β2)) = 0
because this projection is orthogonal.
The i = 3 case is the same as the i = 2 case except for one detail. As in the
i = 2 case, expanding the definition
�κ3 = � β3 − � β3 �κ1
· �κ1 −
�κ1 �κ1
� β3 �κ2
· �κ2
�κ2 �κ2
= � β3 − � β3 �κ1
·
�κ1 �κ1
� β1 − � β3 �κ2
·
�κ2 �κ2
� β2
� − � β2 �κ1
·
�κ1 �κ1
� β1
shows that �κ3 is nonzero and is in the span. A calculation shows that �κ3 is
orthogonal to the preceding vector �κ1.
� �β3 − proj [�κ1]( � β3) − proj [�κ2]( � β3) �
�κ1 �κ3 = �κ1
= �κ1
�
�β3 − proj [�κ1]( � β3) � − �κ1 proj [�κ2]( � =0
β3)
(Here’s the difference from the i = 2 case—the second line has two kinds of
terms. The first term is zero because this projection is orthogonal, as in the
i = 2 case. The second term is zero because �κ1 is orthogonal to �κ2 and so is
orthogonal to any vector in the line spanned by �κ2.) The check that �κ3 is also
orthogonal to the other preceding vector �κ2 is similar. QED
Beyond having the vectors in the basis be orthogonal, we can do more; we
can arrange for each vector to have length one by dividing each by its own length
(we can normalize the lengths).
2.8 Example Normalizing the length of each vector in the orthogonal basis of
Example 2.6 produces this orthonormal basis.
⎛
1/
〈 ⎝
√ 3
1/ √ 3
1/ √ ⎞ ⎛
−1/
⎠ , ⎝
3
√ 6
2/ √ 6
−1/ √ ⎞ ⎛
⎠ , ⎝
6
−1/√2 0
1/ √ ⎞
⎠〉
2
Besides its intuitive appeal, and its analogy with the standard basis En for R n ,
an orthonormal basis also simplifies some computations. See Exercise 17, for
example.
Exercises
2.9 Perform the Gram-Schmidt process on each of these bases for R 2 .
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