MAT225 Section Summary: 6.2

Orthogonal Sets

Summary

orthogonal set: A set of vectors tex2html_wrap_inline343 in tex2html_wrap_inline345 is said to be an orthogonal set if

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In many cases we like our bases to be orthogonal (that is, the vectors to be mutually perpendicular). Even better are orthonormal bases, in which the orthogonal vectors are of unit length.

Theorem 4: If tex2html_wrap_inline347 is an orthogonal set of nonzero vectors in tex2html_wrap_inline345 , then S is linearly independent and hence is a basis for the subspace spanned by S.

orthogonal basis: an orthogonal basis for a subspace W of tex2html_wrap_inline345 is a basis for W that is also an orthogonal set.

Theorem 5: Let tex2html_wrap_inline361 be an orthogonal basis for a subspace W of tex2html_wrap_inline345 . For each tex2html_wrap_inline367 in W, the weights in the linear combination

  equation100

are given by

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Whoops! We have a notational collision: the author wants to use the ``hat'' symbol to indicate the orthogonal projection of y onto another vector. I don't like this, because the vector tex2html_wrap_inline371 is different for different vectors. Mathworld (maintainers of Mathematica), many other mathematicians, and I like to reserve the ``hat'' for unit vectors gif

To make your lives easier, I'll give up my notation, albeit unhappily. I'll indicate unit vectors by the notation tex2html_wrap_inline373 : hence,

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So

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I prefer this right-most form of the projection, as it makes clear what's going on: we form a unit vector tex2html_wrap_inline373 in the direction of u, cast a shadow along this unit vector using the inner product, and then weight the normal vector tex2html_wrap_inline373 by this coefficient. This corresponds to the ``shadow'' cast by the vector tex2html_wrap_inline379 onto the of vector u. Then we can write

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where z is orthogonal to u. We can rewrite equation 1 as

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which just says that we break vector y into its components along the orthogonal direction to represent it. This is what we do with our ordinary basis of vectors tex2html_wrap_inline381 , tex2html_wrap_inline383 , and tex2html_wrap_inline385 .

The fact of the matter is that orthonormal bases are used more often than orthogonal bases, so we generally are working with normalized vectors.

Theorem 6: An tex2html_wrap_inline387 matrix U has orthonormal columns if and only if tex2html_wrap_inline391 .

Theorem 7: Let U be an tex2html_wrap_inline387 matrix with orthonormal columns, and let x and y be in tex2html_wrap_inline345 . Then

  1. tex2html_wrap_inline399
  2. tex2html_wrap_inline401
  3. tex2html_wrap_inline403 if and only if tex2html_wrap_inline405 .
It is easy to verify these properties. Right?!

Exercise #25, p. 393

Orthogonal matrix: a square matrix such that tex2html_wrap_inline407 , having orthonormal columns. It's ironic that the name is ``orthogonal'', rather than ``orthonormal''. Feel free to call such a matrix an orthonormal matrix.

Curiously enough, orthonormal columns in an orthogonal matrix imply that the rows are also orthonormal:

Example: #28, p. 393

...
see http://mathworld.wolfram.com/NormalizedVector.html
 


LONG ANDREW E
Sat Jan 29 21:06:26 EST 2011