lineaarimuunnoksissa

Lineaarimuunnoksissa is a Finnish term that translates to "linear transformations" in English. In mathematics, a linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that for any vectors $u$ and $v$ in the first vector space and any scalar $c$, the following two properties must hold: $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$.

Linear transformations are fundamental in linear algebra and have wide-ranging applications in various fields, including physics, engineering, computer graphics, and economics. They can be represented by matrices. If a linear transformation maps from an $n$-dimensional vector space to an $m$-dimensional vector space, its matrix representation will be an $m \times n$ matrix. Applying the transformation to a vector is equivalent to multiplying the vector by its corresponding matrix.

The set of all linear transformations between two vector spaces forms a vector space itself. Key properties of linear transformations include mapping the zero vector to the zero vector, preserving linear combinations, and their relationship with the concepts of kernel (null space) and image (range). Understanding lineaarimuunnoksissa is crucial for solving systems of linear equations, analyzing data, and modeling various phenomena.