Choose a basis of the vector subspace and extend it to a basis. State dimensions of the vector subspace and the vector space.
$c)$ $W = \{f(x) \mid f(x) \text{ is a polynomial, degree of f is } \le 3 \text{ and } f(1)=0\}$
Here is what I've figured out so far:
The vector space is $V = \{f(x) \mid f(x) \text{ is a polynomial, degree of f is } \le 3\} = P_3$ and it has dimension $n+1=4$.
I'm given that if $f(x) = a_1x^3 + b_1x^2 + c_1x + d_1$ then $f(1) = a_1+b_1+c_1+d_1=0$
Anyone know how to answer this or at least get it started?
Best Answer
You can note that $f(x) = a_1x^3 + b_1x^2 + c_1x + d_1$ then $f(1) = a_1+b_1+c_1+d_1=0$ therefore $ f(x)=a_1(x^3-1)+b_1(x^2-1)+c_1(x-1)$ and a basis is given from
$$x^3-1 ,x^2-1 ,x-1$$