0
$\begingroup$

This question already has an answer here:

are there any theorems that give us any conditions to know if a linear subspace $E$ plus its orthogonal complement span the whole vector space?

For exemple, I know that in $\mathbb{R}[X]$, the complement orthogonal of the hyperspace $Span(1+X, 1+X^2, ...)$ is $\{0\}$ and thus the sum does not span the whole space

Thanks !

$\endgroup$

marked as duplicate by Adrian Keister, Theoretical Economist, Namaste linear-algebra Sep 7 '18 at 0:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 2
    $\begingroup$ In order to define "orthogonal complement" you need an inner product, or at least a bilinear form. Which bilinear form are you using on $\mathbb R[X]$? $\endgroup$ – Robert Israel Sep 6 '18 at 20:38
0
$\begingroup$

Hint: you can prove that $(A + B)^T=A^T \cap B^T$, this is rather easy. Then it is sufficient to notice that $(A^T+A)^T=A^T \cap A=0$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.