Difference between revisions of "Inner Products MA265F10Walther" - Rhea

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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

Back to MA265 Fall 2010 Prof Walther

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