Inner Product Spaces Calculator

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Example

Created on 2024-06-20Asked by Jackson Hernandez (Solvelet student)

Given vectors

\mathbf{u} = (1, 2, 3)

and

\mathbf{v} = (4, 5, 6)

\mathbb{R}^3

, find their inner product.

Solution

To find the inner product of vectors

\mathbf{u} = (1, 2, 3)

and

\mathbf{v} = (4, 5, 6)

\mathbb{R}^3

: The inner product (dot product) of two vectors

\mathbf{u}

and

\mathbf{v}

is given by:

\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3

Substitute the components of

\mathbf{u}

and

\mathbf{v}

\mathbf{u} \cdot \mathbf{v} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6

\mathbf{u} \cdot \mathbf{v} = 4 + 10 + 18

\mathbf{u} \cdot \mathbf{v} = 32

Therefore, the inner product of

\mathbf{u}

and

\mathbf{v}

is:

\mathbf{u} \cdot \mathbf{v} = 32

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Madison Miller on Solvelet

1. Verify whether the inner product

\langle u, v \rangle = u_1 v_1 + u_2 v_2

is valid for the vector space

\mathbb{R}^2

,2. Compute the angle between the vectors

\mathbf{u} = \langle 1, -1, 2 \rangle

and

\mathbf{v} = \langle 3, 4, 5 \rangle

\mathbb{R}^3

using their inner product and norms.,

DefinitionThe complex or real vector spaces for which an inner product(X,Y) is defined in the space so that we can measure lengths and angles are called inner product spaces (alternative termHilbert space). An inner product 〈u,v〉 expected to enjoy properties like: linearity, symmetry, positive-definiteness. Inner product spaces are generalizations of Hilbert spaces and are of great importance in functional analysis, quantum mechanics and signal processing. For Example: In the vector space R2, the inner product of vectors u=(u1,u2) and v=(v1,v2) is given by ⟨u,v⟩=u1v1+u2v2.

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