skalárszorzata

Skalárszorzata, in English scalar multiplication, is a fundamental operation in linear algebra. It involves multiplying a vector by a scalar, which is a single number. The result of this operation is a new vector that is collinear with the original vector. The direction of the new vector is the same as the original if the scalar is positive, and the opposite direction if the scalar is negative. If the scalar is zero, the resulting vector is the zero vector.

The magnitude of the resulting vector is scaled by the absolute value of the scalar. For example, if a vector v is multiplied by a scalar c, the resulting vector cv will have a length that is |c| times the length of v. The components of the vector are each multiplied by the scalar. If v = (v1, v2, ..., vn) and c is a scalar, then cv = (cv1, cv2, ..., cvn). This property allows for the scaling of geometric objects represented by vectors, such as stretching or shrinking them.

Scalar multiplication is a key property that defines vector spaces. It, along with vector addition, satisfies the axioms of a vector space. These axioms ensure that vector operations behave in a predictable and consistent manner. The operation is widely used in various fields including physics, engineering, computer graphics, and machine learning for tasks such as transformations, interpolation, and representing physical quantities.