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 Feb23 comment Prove $\sin^2\theta + \cos^2\theta = 1$ @Sami Ben Romdhane How do we get last implication? Where did it come from? Feb22 answered Find a vector that spans the given set Feb22 comment Find a vector that spans the given set -2,5,3 is a vector from H. I think the answer 1,3,2 is incorrect. How do we get -2,5,3 from 1,3,2? Feb22 comment Basis for a vectorspace The condition given, $\text{dim}(U)=2$, implies they exist Feb22 comment Basis for a vectorspace condition $\text{dim}(U)=2$ implies that $U$ is spanned by two independent vectors from $\mathbb{R}^3$. Task is to show them. Feb22 comment Basis for a vectorspace are you sure this is written as you said? Dec20 awarded Constituent Dec13 comment How prove this rank identity $r(A)=r(B)$ Thank you for correcting my mistaken answer and giving us this more structural proof Dec8 awarded Caucus Dec7 revised Finding a basis for two subspaces of $\mathbb {R}^4$ added 324 characters in body Dec7 answered Finding a basis for two subspaces of $\mathbb {R}^4$ Dec7 awarded Yearling Dec7 revised Prove that if $f'(x)\gt 0$ , then the function is increasing, and if $f'(x)\lt 0$ then the function is decreasing. deleted 60 characters in body Nov2 answered Prove that if $f'(x)\gt 0$ , then the function is increasing, and if $f'(x)\lt 0$ then the function is decreasing. Nov2 answered Linear Independence in R^n Nov2 revised Prove an implication about quadratic form definiteness added 32 characters in body Oct7 comment Prove an implication about quadratic form definiteness @Daniel thank you very much Oct7 comment Prove an implication about quadratic form definiteness @Daniel yes, that's it, you can post this as an answer, but I think we should first show that result is true for n=2, because for n=1 it is trivial. Then we show that it is true for any n, based on fact that n-1 is positive definite. The result for n comes always from the fact that $\det A^TA = ( \det A )^2>0$ Oct7 comment Prove an implication about quadratic form definiteness We want to prove => in statement: quadratic form $q$ over an $n$-dimensional real linear space $V$ is positive definite $\iff$ all main minors $\Delta_1, \Delta_2, ..., \Delta_n > 0$ Oct7 comment Prove an implication about quadratic form definiteness @Daniel "Thus, by induction on the dimension its main minors are positive." - but the fact that if q is positive then minors are positive is what we have to PROVE now