Determine whether the given set of functions is linearly dependent or linearly independent.

Question

a. f₁(t)=2t-3 , f₂(t)=t²+1 , f₃(t)=2t²-t , f₄(t)=t²+t+1

b. f₁(t)=2t-3 , f₂(t)=t²+1 , f₃(t)=2t²-t

Answer 1

b. The set is linearly independent if and only if the only values of A, B, C in the equation

Af₁ + Bf₂ + Cf₃ = 0t² + 0t + 0 are A = B = C = 0.

We have A(2t - 3) + B(t² + 1) + C(2t² - t) = 0t² + 0t + 0

Equating coefficients of like powers, we get B + 2C = 0, 2A - C = 0 and -3A + B = 0

From the second equation, C = 2A.

Replace C by 2A in the equation B + 2C = 0 to get B = -4A.

So, since B = -4A and -3A + B = 0, we have -7A = 0.

Thus, A = 0, B = -4A = 0, and C = 2A = 0

Therefore, the set is linearly independent.

a. The set of all second degree polynomials (in the variable t) with real coefficients is a 3-dimensional vector space with basis {1, t, t²}. Since any subset of a 3-dimensional vector space that contains more than 3 elements is linearly dependent, the given set is NOT linearly independent.