- #1
captainjack2000
- 99
- 0
1. Consider the vector space of polynomials 1+x^3 , 1-x+x^2, 2x, 1+x^2
Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors?
3. I have tried to solve this by letting
a1 = 1+x^3
a2 = 1-x+x^2
a3 = 2x
a4 = 1+x^2
Then I let
(alpha)a1 + (beta)a2 + (gamma)a3 + (delta)a4 = 0
(alpha)(1+x^3) + (beta)(1-x+x^2) + (gamma)(2x) +(delta)(1+x^2) = 0
(alpha +beta+delta) + x(2gamma - beta) + x^2(beta + delta) + x^3(alpha) = 0
So alpha + beta+ delta = 0
2gamma - beta = 0
beta + delta = 0
alpha = 0
But I can't solve for beta delta and gamma so how do I know if their independent or dependent?
Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors?
3. I have tried to solve this by letting
a1 = 1+x^3
a2 = 1-x+x^2
a3 = 2x
a4 = 1+x^2
Then I let
(alpha)a1 + (beta)a2 + (gamma)a3 + (delta)a4 = 0
(alpha)(1+x^3) + (beta)(1-x+x^2) + (gamma)(2x) +(delta)(1+x^2) = 0
(alpha +beta+delta) + x(2gamma - beta) + x^2(beta + delta) + x^3(alpha) = 0
So alpha + beta+ delta = 0
2gamma - beta = 0
beta + delta = 0
alpha = 0
But I can't solve for beta delta and gamma so how do I know if their independent or dependent?