Skalárszorzatra

Skalárszorzatra, which translates to scalar multiplication in Hungarian, is a fundamental operation in linear algebra. It involves multiplying a vector by a scalar, which is a real or complex number. The result of this operation is another vector that has the same direction as the original vector but a different magnitude, or a reversed direction if the scalar is negative.

If a vector v is multiplied by a scalar c, the resulting vector cv has components that are each multiplied by c. For example, if v = [v1, v2] and c is a scalar, then cv = [c*v1, c*v2]. The length of the resulting vector is the absolute value of the scalar multiplied by the length of the original vector. If the scalar is 1, the vector remains unchanged. If the scalar is 0, the result is the zero vector.

Scalar multiplication has important properties, including distributivity over vector addition (c(u + v) = cu + cv) and associativity with scalar multiplication ((cd)v = c(dv)). It is a key concept used in defining many other vector operations and in understanding geometric transformations like scaling. This operation is crucial in various fields, including physics, computer graphics, and engineering, wherever quantities are described by both magnitude and direction.

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