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suko123 December 7th, 2014 04:27 PM

Inner Product
 
Are the standard inner product and inner product the same thing? If not please give examples regarding two matrix vectors.

Prokhartchin December 8th, 2014 03:59 AM

DEFINITION:

See the following definition here.

Let a vector space $V$ over the field $F$. An inner product is a map

$$\langle \cdot, \cdot \rangle : V \times V \to F$$

that satisfies the following three axioms for all vectors $x,y,z \in V$ and all scalars $a \in F$:

1) Conjugate symmetry:

$$\langle x,y\rangle =\overline{\langle y,x\rangle}.$$

Note that when $F = \mathbb{R}$, conjugate symmetry reduces to symmetry. That is, $\langle x,y \rangle = \langle y,x \rangle$ for $F = \mathbb{R}$; while for $F = \mathbb{C}, \langle x,y \rangle$ is equal to the complex conjugate of the number $\langle y,x \rangle$.

2) Linearity in the first argument:

$$ \langle ax,y\rangle = a \langle x,y\rangle, \\
\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle.$$

Together with conjugate symmetry, this implies conjugate linearity in the second argument (below).

3) Positive-definiteness:

$$\langle x,x\rangle \geq 0
\langle x,x\rangle = 0 \Rightarrow x = 0$$
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EXAMPLES:

A) If $V=\mathbb{R}^2$ the standard (canonical, simplest) inner product of two vector $v=(x_1,x_2), u=(y_1,y_2) \in V$ is defined by
$$\langle \cdot, \cdot \rangle : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}, \\
\langle (x_1,x_2), (y_1,y_2) \rangle = x_1y_1 + x_2y_2.$$

For confirm this you should verify that this map satisfies 1), 2) and 3) of definition of inner product.


B) If $V=C([a,b])=$"continuous functions defined in" $[a,b] = \{f:[a,b]\rightarrow\mathbb{R}/ $ f is a continuous function$\}$ the standard (canonical, simplest) inner product of two vector $v=f(t), u=g(t) \in V$ is definde by
$$\langle \cdot, \cdot \rangle : C([a,b])\times C([a,b]) \to \mathbb{R}, \\
\langle f, g \rangle =\int_a^b f(t)g(t) dt$$

For confirm this you should verify that this map satisfies 1), 2) and 3) of definition of inner product.

C) If $V=\mathbb{R}^2$ the you can define the inner product of two vector $v=(x_1,x_2), u=(y_1,y_2) \in V$ by
$$\langle \cdot, \cdot \rangle : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}, \\
\langle (x_1,x_2), (y_1,y_2) \rangle := x_1y_1 - x_1y_2-x_2y_1+3x_2y_2.$$

For confirm this you should verify that this map satisfies 1), 2) and 3) of definition of inner product.

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In example A) you have the standard inner product when the vector space is the Euclidean space $\mathbb{R}^2$. In C) you have a inner product more complicated for this vector space ($V=\mathbb{R}^2$). If you to try check 1),2) and 3) in C) you see that is indeed more complicated.

In B) you see a vector space of functions. In this space the more simple and usual map that is a inner product is that integral.

In this tree examples note that you take two elements of a vector space (called vectors) and mapping these (by the inner product) in scalar field, obtained a scalar (a number). Exist different maps that permit you realize this. Some are more simple, other more complicated.


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