Let $V$ and $W$ be vector spaces over $K$ and $f: V \to W$ a Homomorphism. Prove that this is a Homomorphism: $f^: V^ \to W^*, g \to g \circ f$

51 Views Asked by Bumbble Comm At 30 Mar 2026 - 1:54

Let $V$ and $W$ be vector spaces over $K$ and $f: V \to W$ a Homomorphism. Prove that this is a Homomorphism. $$f^*: V^* \to W^*, g \to g \circ f$$

My attempt:

Homomorphism means $f(\alpha v + w) = \alpha f(v) + f(w)$

And we have

$g \circ f = g(f(x))$

Somehow we must get

$g \circ f(\alpha v + w) = \alpha (g \circ f)(v) + (g \circ f)(w)$

but how ? Any hint would be welcome.

Let $V$ and $W$ be vector spaces over $K$ and $f: V \to W$ a Homomorphism. Prove that this is a Homomorphism: $f^*: V^* \to W^*, g \to g \circ f$