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).

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

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 $f$.


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