Question

Determine whether S is a basis for P_3:
S={4t-t^2, 5+t^3, 3t+5, 2t^3-3t^2}

Answer 1

what that

Answer 2

let me guess a linear equation

Answer 3

i think...
[0 -1 4 0]
[1 0 0 5]
[0 0 3 5]
[2 -3 0 0].

Answer 4

a(- t^2 + 4t) + b(t^3 + 5) + c(3t + 5) + d(2t^3 - 3t^2)

Answer 5

understand?

Answer 6

There are two conditions for a set of vectors to be a basis. One: They must be linearly independent, and two: they must span the set.
Lets see if they are linearly independent:
\[\lambda (4t-t^2)+\gamma (5+t^3)+\rho (3t+5)+\phi(2t^3-3t^2)=0\]
So now we need to see that this ONLY holds for lambda=gamma=rho=phi=0.
So:
\[(4\lambda+3\rho)t+(-\lambda-3\phi)t^2+(\gamma+2 \phi)t^3+5(\gamma+\rho)=0\]
So
\[4 \lambda=-3 \rho; -\lambda=3 \phi; \gamma=-2 \phi; \gamma=-\rho\]
So you get:
\[\lambda=-3 \phi; \rho=2\phi; \rightarrow 4\lambda=-3\rho \implies \rho=\phi=\lambda=\gamma=0\]

Answer 7

Now, does it span the space?

Answer 8

wait i am just reading what u wrote =)

Answer 9

it is enough to show that the determinant of fontez's matrix is non zero

Answer 10

yes it spans =)

Answer 11

it is basis for P_3

Answer 12

put your hands in the air

Answer 13

NOW DANCE!!!!!!!!

Answer 14

get down

Answer 15

hehe I am

Answer 16

Thanks maelvolence

Answer 17

POLICE PUT YOUR HAND UP