A First Course in Linear Algebra, 2017a

238 R n
To show that {⃗v 1 ,···,⃗v n } is an orthogonal set, let
a 2 = ⃗u 2 •⃗v 1
‖⃗v 1 ‖ 2
then:
⃗v 1 •⃗v 2 =⃗v 1 • (⃗u 2 − a 2 ⃗v 1 )
=⃗v 1 •⃗u 2 − a 2 (⃗v 1 •⃗v 1
=⃗v 1 •⃗u 2 − ⃗u 2 •⃗v 1
‖⃗v 1 ‖ 2 ‖⃗v 1‖ 2
=(⃗v 1 •⃗u 2 ) − (⃗u 2 •⃗v 1 )=0
Now that you have shown that {⃗v 1 ,⃗v 2 } is orthogonal, use the same method as above to show that {⃗v 1 ,⃗v 2 ,⃗v 3 }
is also orthogonal, and so on.
Then in a similar fashion you show that span{⃗u 1 ,···,⃗u n } = span{⃗v 1 ,···,⃗v n }.
Finally defining ⃗w i = ⃗v i
for i = 1,···,n does not affect orthogonality and yields vectors of length 1,
‖⃗v i ‖
hence an orthonormal set. You can also observe that it does not affect the span either and the proof would
be complete.
♠
Consider the following example.
Example 4.136: Find Orthonormal Set with Same Span
Consider the set of vectors {⃗u 1 ,⃗u 2 } given as in Example 4.117. Thatis
⎡ ⎤ ⎡ ⎤
1 3
⃗u 1 = ⎣ 1 ⎦,⃗u 2 = ⎣ 2 ⎦ ∈ R 3
0 0
Use the Gram-Schmidt algorithm to find an orthonormal set of vectors {⃗w 1 ,⃗w 2 } having the same
span.
Solution. We already remarked that the set of vectors in {⃗u 1 ,⃗u 2 } is linearly independent, so we can proceed
with the Gram-Schmidt algorithm:
⎡ ⎤
1
⃗v 1 = ⃗u 1 = ⎣ 1 ⎦
0
( ) ⃗u2 •⃗v 1
⃗v 2 = ⃗u 2 −
‖⃗v 1 ‖ 2 ⃗v 1
=
=
⎡
⎣
⎡
⎢
⎣
3
2
0
⎤
⎡
⎦ − 5 ⎣
2
1
2
− 1 2
0
⎤
⎥
⎦
1
1
0
⎤
⎦