Question

Determine whether the given set of vectors in

M^2 (R) is linearly independent. a1 ,a2,a3 where ai are vectors in M^2(R).
a) Yes, linearly independent
b) No, linearly dependent

Answer 1

To determine whether the given set of vectors in M^2 (R) is linearly independent, we need to check if the coefficient of each vector in a linear combination is equal to zero only when all the coefficients are zero.

Step-by-step explanation:

To determine whether the given set of vectors in M^2 (R) is linearly independent, we need to check if the coefficient of each vector in a linear combination is equal to zero only when all the coefficients are zero.

Let's say the given set of vectors is {a1, a2, a3}. If we can find coefficients x1, x2, and x3 (not all zero) such that x1*a1 + x2*a2 + x3*a3 = 0, then the vectors are linearly dependent. Otherwise, they are linearly independent.

For example, if we have a1 = (1, 0), a2 = (0, 1), and a3 = (1, 1), we can see that there are no coefficients x1, x2, and x3, not all zero, that satisfy the equation x1*a1 + x2*a2 + x3*a3 = 0. Therefore, the vectors {a1, a2, a3} are linearly independent.