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Zero-product property. In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a singular matrix where the determinant is 0. In this process, information is lost and cannot be regained. For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
Multiplying by 0 gives the zero vector: 0v = 0; Multiplying by −1 gives the additive inverse: (−1)v = −v. Here, + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field. Interpretation
If the cross product of two vectors is the zero vector (that is, a Ă— b = 0), then either one or both of the inputs is the zero vector, (a = 0 or b = 0) or else they are parallel or antiparallel (a ∥ b) so that the sine of the angle between them is zero (θ = 0° or θ = 180° and sin θ = 0). The self cross product of a vector is the zero ...
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
As is the case with addition, ordinal multiplication on the natural numbers is the same as standard multiplication. Properties. α · 0 = 0 · α = 0, and the zero-product property holds: α · β = 0 → α = 0 or β = 0. The ordinal 1 is a multiplicative identity, α · 1 = 1 · α = α. Multiplication is associative, (α · β) · γ = α ...
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