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Revision as of 09:05, 8 December 2010
Inner Product Spaces and Orthogonal Complements
Introduction
The following entries are derived from a relatively large yet concise topic called Inner Product Spaces. I would focus on two subtopics which are the Inner Product Spaces themselves and Orthogonal Complements. Other essential subtopics would also be posted in the form of background knowledge to ensure the thoroughness of readers' understanding. Please also note that the Cross Products subtopic is not required in the context of MA 26500.
Part 1: Inner Product Spaces
Background Knowledge - The Basics of Vectors
Inner Product Spaces
As appeared briefly for calculating the angle between two vectors in the background knowledge, the standard scalar inner product is defined as:
- $ \langle v,w\rangle = v_1w_1 + v_2w_2 + \cdots + v_nw_n $
for any vectors v and w in Rn.
The properties of this standard inner product are:
- $ \langle v,v\rangle \geq 0 $; $ \langle v,v\rangle = 0 $
- if and only if v = 0;
- $ \langle v,w\rangle = \langle w,v\rangle $;
- $ \langle v + u,w\rangle = \langle v,w\rangle + \langle u,w\rangle $;
- $ \langle kv,w\rangle = \langle v,kw\rangle = k\langle v,w\rangle $.
Another type of inner products can be seen in continuous functions such as in the form of:
- $ \langle f,g\rangle = \int_a^b \! f(x)g(x)\,dx \, $
where x is greater than zero.
In continuation, an inner product space is a vector space with an inner product. Orthogonality in an inner product space occurs when the following example of conditions occurs:
- $ \langle v,w\rangle = 0 $
- and
- $ \langle v,v\rangle = 1 $
- and
- $ \langle w,w\rangle = 1 $.
Lastly, the Cauchy-Bunyakovsky-Schwarz (CBS) Inequality states the following for inner product spaces:
- $ |\langle v,w\rangle| \leq \|v\| \|w\|\, $.
In most contexts, this is indeed equivalent to the Triangle Inequality that states:
- $ \displaystyle \|v + w\| \leq \|v\| + \|w\| $
where v and w are the shorter vectors of a triangle.
Part 2: Orthogonal Complements
Background Knowledge - The Gram-Schmidt Algorithm
Orthogonal Complements
Hint - The Least Squares Solution
Ryan Jason Tedjasukmana