Section 5.1 Inner Products
Recall the dot product of \(\vec{v} =\colvec{v_1\\ v_2\\ \vdots\\
v_n} \in \mathbb{R}^n\) and \(\vec{w} =\colvec{w_1\\ w_2\\ \vdots\\
w_n} \in \mathbb{R}^n\) is the sum of the products of the components. Namely,
\begin{equation*}
\vec{v} \cdot \vec{w} =\sum_{i=1}^n v_i w_i =\vec{v}^T \vec{w}\text{.}
\end{equation*}
The dot product of a vector \(\vec{x} \in \mathbb{R}^n\) with itself gives the length of the vector squared, \(\vec{x} \cdot \vec{x} = \|\vec{x}\|^2\text{.}\) The dot product is the familiar example of an inner product on a real vector space.
If \(z = a+b i \in \mathbb{C}\text{,}\) the conjugate of \(z\) is denoted \(\overline{z}\) and computed as \(\overline{z} = a -b
i \text{.}\)
Definition 5.1.
An inner product on a vector space \(V\) is a function from \(V \times V\) to \(\mathbb{R}\) for real vector spaces (\(\mathbb{C}\) for complex vector spaces), denoted by \(\langle *,*\rangle\text{,}\) such that for all \(\vec{x},\vec{y},\vec{z} \in V\) and \(c \in \mathbb{R}\) (or \(\mathbb{C}\)):
\(\langle \vec{x},\vec{y} \rangle=\langle \vec{y},\vec{x}
\rangle\) (or \(\langle \vec{x},\vec{y} \rangle=\overline{\langle
\vec{y},\vec{x} \rangle}\) for \(\mathbb{C}\))
\(c \langle \vec{x},\vec{y} \rangle=\langle c\vec{x},\vec{y} \rangle\) and \(\langle \vec{x}+\vec{z},\vec{y} \rangle=\langle \vec{x},\vec{y} \rangle+\langle \vec{z},\vec{y} \rangle\)
\(\displaystyle \langle \vec{x},\vec{x} \rangle \geq 0\)
A vector space with a defined inner product is called an inner product space.
Example 5.2.
\(\mathbb{R}^n\) with the dot product defined above is an inner product space.
\(C([0,1])\text{,}\) the set of continuous functions on the interval
\([0,1]\text{,}\) is an inner product space when
\begin{equation*}
\langle f,g \rangle = \int_0^1 f(t)g(t) \, dt
\end{equation*}
Frobeinus Inner Product on Matrices: If
\(A,B \in M_{m
\times n}(\mathbb{R})\text{,}\) then
\begin{equation*}
\langle A,B \rangle =
\sum_{i=1}^{m}\sum_{j=1}^{n} A_{i,j} B_{i,j}
\end{equation*}
is an inner product on
\(M_{m \times n}(\mathbb{R})\text{.}\)
Definition 5.3.
Two non-zero vectors \(\vec{x}\) and \(\vec{y}\) in an inner product space are orthogonal if \(\langle \vec{x}
,\vec{y} \rangle=0\text{.}\)
Question 5.1.
Find 3 different vectors in \(\mathbb{R}^2\) that are orthogonal to \(\colvec{1\\ 2}\text{.}\)
Exercise 5.4.
Find 3 different vectors in \(\mathbb{R}^3\) that are orthogonal to \(\colvec{1\\ 2\\ -1}\text{.}\)
Question 5.2.
Find a nonzero vector in \(C([0,1])\) that is orthogonal to \(f(t)=1\text{.}\)
Question 5.3.
Find a nonzero vector in \(C([0,1])\) that is orthogonal to \(f(t)=t\text{.}\)
Exercise 5.5.
Find a nonzero vector in \(M_{2 \times
3}(\mathbb{R})\) that is orthogonal to \(A=\begin{bmatrix}0\amp 2\amp 1\\1\amp -3\amp 0
\end{bmatrix}\text{.}\)
Definition 5.6.
For vectors in \(\mathbb{R}^n\text{,}\) the projection of \(\vec{x}\) onto \(\vec{y}\) computed with the following:
\begin{equation*}
\proj_{\vec{y}} \vec{x} = \left( \frac{\vec{x} \cdot
\vec{y}}{\vec{y} \cdot \vec{y}} \right) \vec{y}\text{.}
\end{equation*}
Question 5.4.
(a)
Compute \(\proj_{\vec{u}}\vec{v}\) with \(\vec{u} = \langle 2,2 \rangle\) and \(\vec{v} = \langle 1,3
\rangle\text{.}\)
(b)
Plot \(\vec{u}\text{,}\) \(\vec{v}\text{,}\) and \(\proj_{\vec{u}} \vec{v}\) starting at the origin.
(c)
Write a few sentences about what the projection measures geometrically.
Inner product spaces are useful because the same argument we made in the previous problem about how much of one vector is in the direction of another can be generalized to vector spaces that do not have the geometric interpretation of arrows in space.
