# How is a linear transformation defined?

Let $V$ and $W$ be vector spaces over the same field $K$. A function $f : V \to W$ is said to be a linear map if for any two vectors $x$ and $y$ in $v$ and any scalar $\alpha$ in $K$, if it satisfies the condition that the image of a linear combination of vectors in $V$ is the linear combination of images of the said vectors.

Now, what if the rules for scalar multiplication and addition in the two vector fields are different? Will the evaluation of the condition's truth involve use of both the definitions?

• Yes, the addition and multiplication by scalar have to be transformed to the new addition and multiplication by scalar in the new space. – KittyL Feb 8 '15 at 10:17

To say that $f:V \to W$ is linear means that if $x,y \in V$ and $\alpha, \beta \in K$ then $T(\alpha \cdot x + \beta \cdot y) = \alpha \odot T(x) \oplus \beta \odot T(y)$. Here, $\cdot$ and $+$ denote the scalar multiplication and addition operations for $V$, and $\odot$ and $\oplus$ denote the scalar multiplication and addition operations for $W$.

Usually people use the same symbol for both $+$ and $\oplus$, however. And they typically omit the symbols $\cdot$ and $\odot$ entirely.

A linear map $f:V\rightarrow W$ by definition satisfies

$\forall v_1,v_2\in V\quad f(v_1+v_2)=f(v_1)+f(v_2)$

and

$\forall \lambda\in K \; v\in V\quad f(\lambda\cdot v)=\lambda\cdot f(v)$.

Here the $+$ and $\cdot$ on the left sides of the equations in general are different operations than the $+$ and $\cdot$ on the right sides.