# linearity of transformation

Prove that a function $$f: V \rightarrow \mathbf{R}^{m}$$ is linear if and only if each component of $$f$$ is linear. Determine the form of a linear map $$T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m} .$$

I get the the form of a linear map $$T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m} .$$ which is consequence of first part. But how to show first part for arbitrary vector space (infinite dimension may work).

• What form did you get for $T$? – Berci Oct 23 '19 at 7:29
• @Berci It is $m \times n$ matrix A such that it takes vector x and send Ax to $R^m$ . – maths student Oct 23 '19 at 7:35

## 1 Answer

Let $$p_i:\Bbb R^n\to\Bbb R$$ be the projection to the $$i$$th coordinate. This is clearly linear, as all vector operations in $$\Bbb R^n$$ are defined coordinatewise.
Thus, if $$f:V\to \Bbb R^n$$ is linear, so are the compositions $$p_i\circ f$$.

For the converse, we assume that $$f_i=p_i\circ f$$ is linear for each index $$i$$, and we have $$f(v) =\pmatrix{f_1(v)\\ \vdots\\f_n(v)}$$ so, again using the componentwise property of operations, one can directly prove linearity of $$f$$.