inner product

/'inə'prɔdʌkt/
Học thuật
Thân thiện
inner product

A student calculates the inner product of two vectors on a chalkboard.

Definition
  1. Noun:
    • A scalar quantity resulting from the multiplication of two vectors: In mathematics, particularly linear algebra, the inner product is a fundamental operation that takes two equal-length sequences of numbers (vectors) and returns a single number (a scalar). This scalar provides a measure of the vectors' magnitude alignment and the angle between them. It generalizes the dot product to various vector spaces.
Examples of Usage
  • Noun:
    • The inner product of two perpendicular vectors is zero.
    • Calculating the inner product is a crucial step in determining the angle between two vectors.
    • In quantum mechanics, the inner product is used to define the probability amplitude.
Advanced Usage
  • "To compute/take the inner product": To perform the mathematical operation of finding the inner product.
    • The algorithm requires us to compute the inner product of these two high-dimensional vectors.
  • "Inner product space": A vector space equipped with an inner product operation, which allows for the definition of geometric concepts like length and angle.
    • Hilbert spaces are a specific type of complete inner product space.
Variants and Related Words
  • Dot product (n): Often used synonymously with inner product in the context of Euclidean space (ℝⁿ). It is the standard inner product for coordinate vectors.
    • For vectors in three-dimensional space, the inner product is typically the dot product.
  • Scalar product (n): Another common synonym emphasizing that the result is a scalar, not a vector.
  • Hermitian inner product (n): A type of inner product for complex vector spaces, where the result is a complex number, and the operation is conjugate symmetric.
Synonyms
  • Dot product: (Common in Euclidean geometry and physics).
  • Scalar product: (Emphasizes the nature of the result).
Related Phrases
  • Inner product of functions: The concept extended to function spaces, where the "vectors" are functions, and the inner product is often defined as an integral.
    • The inner product of two sine waves over one period can reveal if they are orthogonal.
inner product

A student calculates the inner product of two vectors on a chalkboard.

Noun
  1. a real number (a scalar) that is the product of two vectors

Từ đồng nghĩa