Recent content by Kronos1

You assume that I have not tried a textbook already. coming to this forum is a last resort as I am struggling to understand the definitions in the textbooks. For all the reading I have done I have only ascertained that the basis is a set of vectors that you can make all other vectors in that...

you are correct it is $$D:V \to V, \space y \to \d{y}{x}$$ so far I have set $${e}_{1}\space and\space {e}_{2}=V$$ to prove that the definition of $$f^{\prime\prime}+f=0$$ which was true for both of them. Now to my knowledge to prove a basis I need to prove that they are linearly independent...

The set of all solutions of the differential equation $$\d{^2{y}}{{x}^2}+y=0$$ is a real vector space $$V=\left\{f:R\to R \mid f^{\prime\prime}+f=0\right\}$$ show that $$\left\{{e}_{1},{e}_{2}\right\}$$ is a basis for $V$, where $${e}_{1}:R \to R, \space x \to \sin(x)$$ $${e}_{2}:R \to R...

Hi All struggling with concepts involved here So I have $${P}_{2} = \left\{ a{t}^{2}+bt+c \mid a,b,c\epsilon R\right\}$$ is a real vector space with respect to the usual addition of polynomials and multiplication of a polynomial by a constant. I need to show that both...

Hi Guys having a bit of trouble understanding vector basis. If $$\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$$ is a basis for vector space $V$ over the field $F$ and $${f}_{1}=-{e}_{1}, {f}_{2}={e}_{1}-{e}_{2}, {f}_{3}={e}_{1}-{e}_{3}$$ how can I go about proving that...