My question is
Determine whether the polynomial $p(t) = t^3-3t^2+5t+1, q(t) = t^3-t^2+8t+2, r(t) = 2t^3-4t^2+9t+5$ in the vector space of all polynomials of degree less than or equal to 3 are linear independent.
Following show my try:
How to show " all polynomials of degree less than or equal to 3 are linear independent" this part
Best Answer
First of all, your work is correct and you showed that the $3$ polynomials $p,q,r$ are linearly independent. But the second part that you wanted to show that "all polynomials of degree less than or equal to 3 are linear independent" cannot be true. Look at the example $p = t^3, q = t^2, r = t^3+t^2 \implies -r + p + q = 0\implies p,q, r$ are linearly dependent. But all polynomials of degree less than or equal to $3$ form a vector space.