Finding a 3rd polynomial to create a basis.

[Harambe1]

Hi,

I am struggling with the following problem:

"Let $V=P_3(\Bbb{R})$ and let $t_1=3x^3-x-2$ and $t_2=x^3-3x+2$ with $T=\left\{ t\in V \:|\: t(1)=0 \right\}$. Find ${t_3}\in\left\{T\right\}$ such that $\left\{t_1, t_2, t_2\right\}$ is a basis of T.

Not sure where to go as each column matrix will have 4 elements but then there is only 3 polynomials in the basis.Thanks.

 

[I like Serena]

Hi,

I am struggling with the following problem:

"Let $V=P_3(\Bbb{R})$ and let $t_1=3x^3-x-2$ and $t_2=x^3-3x+2$ with $T=\left\{ t\in V \:|\: t(1)=0 \right\}$. Find ${t_3}\in\left\{T\right\}$ such that $\left\{t_1, t_2, t_2\right\}$ is a basis of T.

Not sure where to go as each column matrix will have 4 elements but then there is only 3 polynomials in the basis.Thanks.

Hi Harambe! Welcome to MHB! ;)

Indeed, there are 4 elements, so a basis of $V$ will have 4 polynomials.
However, $T$ has the restriction $t(1)=0$, meaning we will be left with a basis of 3 polynomials.

Do you have any ideas how we might find that 3rd polynomial?