A First Course in Linear Algebra, 2017a

252 R n −1 1 2 3 4 5
6
y
4
2
x
Regression Line
Data Points
One could clearly do a least squares fit for curves of the form y = ax 2 + bx + c in the same way. In this
case you want to solve as well as possible for a,b, andc the system
⎡ ⎤
x 2 ⎡ ⎤ ⎡ ⎤
1
x 1 1 a y 1
⎢ ⎥
⎣ . . . ⎦⎣
b ⎦ ⎢ ⎥
= ⎣ . ⎦
x 2 n x n 1 c y n
and one would use the same technique as above. Many other similar problems are important, including
many in higher dimensions and they are all solved the same way.
Exercises
Exercise 4.11.1 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine
whether it is also orthonormal.
⎡ √
1
⎤ ⎡ ⎤ ⎡
6√
2
√ 3
1
2√
2 −
⎣ 1
3√ 1 ⎤
3√
3
2 3
− 1 √ √
⎦, ⎣
√
0 ⎦, ⎣ 1
3√
3 ⎦
√
6 2 3
1
2 2
1
3 3
If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the
same span.
Exercise 4.11.2 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine
whether it is also orthonormal.
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 1 −1
⎣ 2 ⎦, ⎣ 0 ⎦, ⎣ 1 ⎦
−1 1 1
If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the
same span.
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