Consider a linear transformation T() = Ax from R2 to R. Suppose for two vectors ūj and vz in R? we have T(1) = 301 and T(02) = 402. What can you say about det A? Justify your answer carefully.

Find the coordinate vector [x]; of x relative to the given basis B= (b1,b2}. 13 -3 X = - 5 b1 b2 -2 - 1 [x]s

Suppose T: R2 R3 is a linear transformation. Let u, v and w be the vectors given below, and suppose that T(u) and T(v) are as given. Find T(w). 18 22 T(u) = 20 25 -7 -8 u= 5 2 V = 0 T(w) = 0 0 5 3 W = -20 10 T(v) =

Consider the vectors i and i shown below, which are related by a reflection through the x-axis. y Suppose T: R' → R? is a linear transformation such that T(e1) = ei and T(7) = 2e1. Find T(w) or explain why it cannot be determined. Warning: The picture does not indicate any scale on the y-axis. That means we cannot estimate the y-components of these vectors j and w (aside from the fact that their y-components are clearly not zero)! 12