Kardi Teknomo
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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Inner Product

By Kardi Teknomo, PhD.

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Vector inner product is also called dot product denoted by Inner ProductorInner Product. Vector inner product is also called vector scalar product because the result of the vector multiplication is a scalar. Vector inner product is closely related to matrix multiplication. It can only be performed for two vectors of the same size.

Geometrically, vector inner product measures the cosine angle between the two input vectors.

Inner Product

Algebraically, the vector inner product is a multiplication of a row vector by a column vector Inner Productto obtain a real value scalar provided by formula below
Inner Product
Some literature also use symbol Inner Productto indicate vector inner product because the in the computation, we only perform sum product of the corresponding element and the transpose operator does not really matter. In this linear algebra tutorial, I use notation Inner Productfor vector inner product to distinct the inner product from the outer product.

Suppose we have Inner ProductandInner Product, the vector inner product is
Inner Product

Try the interactive program of Vector Inner Product below. To use the program, simply click the “Vector Inner Product” button. “Random example” button will give you unlimited examples of the vectors in the right format. You can type your own input vectors.

vector x vector y


Some important properties of vector inner product are

  • Vector inner product is a commutative operation. If you reverse the order you will get the same result but you should notice the transpose operator Inner Product
  • Vector inner product is a distributive operation. You can distribute (and group) the first vector with respect to addition or subtractionInner Product 
  • Vector inner product is an associative operation with respect to scalar multiple of vector. You can exchange the order of computation (operation inside parentheses are to be computed first) does not change the resultInner Product.
  • Vector dot product to itself always produces positive number except when it is a zero vectorInner Product. It produces zero if and only if the vector input is a zero vector.
  • Square of Euclidean norm of a vector is equal to the inner product to itself Inner Product
  • Cosine angle between two vectors is equal to their dot product divided by the product of their normsInner Product
  • Two vectors are perpendicular (orthogonal) to each other if and only if their inner product is zero Inner Product.
  • Dot product of the same standard unit vector is oneInner Product
  • Dot product of the perpendicular standard unit vector is zeroInner Product
  • Relationship of norm of cross product and dot product isInner Product.

See also: Cross product, Outer Product

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\


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