EXAMPLE 3 Using a Matrix for a Linear Transformation For the linear transformation T: R2>R2 from Example 2, find [V]B, [T(v)]3, and [T(v)]B. for the vector v whose coordinate matrix is [V]B: = [3 1]T. SOLUTION To find [V]B, use the transition matrix P from B' to B. Mama aim] To find [T(v)]3, multiply [v]3 on the left by the matrix A to obtain [Tons = MB 2 [:3 gill] : [:3] To find [T(v)]B., multiply [T(v)]3 on the left by P'1 to obtain [mug = P'[T(v)]B = [3 ij : [3] or multiply [V]B. on the left by A' to obtain [mm = Area = [j Hf] = H]