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 only 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
Let
- $ \mathbf{v} = \begin{bmatrix} a \\ b \end{bmatrix} $
which denotes a vector between a point and the origin.
Then the length of this vector is given by
- $ \mathbf{\|v\|} = \sqrt{x^2 +y^2} $.
The same concept applies for two-point case between points v and w as shown below.
Let
- $ \mathbf{w} = \begin{bmatrix} c \\ d \end{bmatrix} $.
We have,
- $ \mathbf{v-w} = \begin{bmatrix} a-c \\ b-d \end{bmatrix} $,
and
- $ \mathbf{\|v-w\|} = \sqrt{(a-c)^2 +(b-d)^2} $.
In short, there is no significant difference in the three dimensional approach of the form
- $ \mathbf{v} = \begin{bmatrix} p \\ q \\ r \end{bmatrix} $.
Another essential concept is the angle between two vectors as shown in the following formula:
- $ \cos{\alpha} = \frac{\langle\mathbf{v}\, , \mathbf{w}\rangle}{\|\mathbf{v}\| \, \|\mathbf{w}\|} $.
This will be more prominent as we go through Inner Product Spaces section.
Lastly, a unit vector is a vector that has magnitude one and denoted as in the following:
- $ \boldsymbol{\hat{w}} = \frac{\boldsymbol{w}}{\|\boldsymbol{w}\|} $.
Inner Product Spaces
Part 2: Orthogonal Complements
Background Knowledge - Gram-Schmidt Algorithm
In the simplest review, the Gram-Schmidt Algorithm is shown in the following pattern for vectors u and v.
- $ \begin{align} \mathbf{u}_1 & = \mathbf{v}_1, \\ \mathbf{u}_2 & = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_2), \\ \mathbf{u}_3 & = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_3)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_3), \\ & {}\ \ \vdots \\ \mathbf{u}_k & = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,(\mathbf{v}_k), \end{align} $
where
- $ \mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} $.
Orthogonal Complements
Part 3: Applications
Generic Homework Problems
Generic Exam Problems
Ryan Jason Tedjasukmana
