Question

If T:R²→R² is a linear transformation show that T(u1) and T(u2) are linearly independent

Answer 1

To show that T(u1) and T(u2) are linearly independent, we need to prove that if a1T(u1) + a2T(u2) = 0, then a1 = 0 and a2 = 0. Using the linearity of T, we can rewrite this equation and conclude that T(u1) and T(u2) are linearly independent.

Step-by-step explanation:

To show that T(u1) and T(u2) are linearly independent, we need to prove that if a1T(u1) + a2T(u2) = 0, then a1 = 0 and a2 = 0.

Let's assume that a1T(u1) + a2T(u2) = 0, where T(u1) and T(u2) are linearly dependent. This means that there exist coefficients a1 and a2, not both zero, such that a1T(u1) + a2T(u2) = 0.

Using the linearity of T, we can rewrite this equation as T(a1u1 + a2u2) = 0. Since T is a linear transformation, T(0) = 0, so we have T(0) = T(a1u1 + a2u2), which implies a1u1 + a2u2 = 0. Since u1 and u2 are linearly independent, a1 = a2 = 0. Therefore, T(u1) and T(u2) are linearly independent.