Show that the transformation T:R^2->R^2 given T(x1,x2)=(3x1-5x2,x1+2x2)
is linear by verifying that satisfies the definition of linear.
To show that the transformation T:R^2 -> R^2 given by T(x1,x2) = (3x1 - 5x2, x1 + 2x2) is linear, we need to verify that it satisfies the definition of linearity.
A transformation T: V -> W is said to be linear if it satisfies two conditions:
1. T(u + v) = T(u) + T(v) for all vectors u and v in V (additivity)
2. T(cu) = cT(u) for all vectors u in V and all scalars c (homogeneity)
Let's check both conditions for our given transformation T(x1,x2) = (3x1 - 5x2, x1 + 2x2):
1. Additivity:
Consider two vectors u = (x1, x2) and v = (y1, y2) in R^2. We need to show that T(u + v) = T(u) + T(v).
T(u + v) = T(x1 + y1, x2 + y2) = (3(x1 + y1) - 5(x2 + y2), (x1 + y1) + 2(x2 + y2))
= (3x1 - 5x2 + 3y1 - 5y2, x1 + x2 + y1 + y2)
= (3x1 - 5x2, x1 + x2) + (3y1 - 5y2, y1 + y2)
= T(x1, x2) + T(y1, y2)
Hence, T satisfies the additivity condition.
2. Homogeneity:
Consider a vector u = (x1, x2) in R^2 and a scalar c. We need to show that T(cu) = cT(u).
T(cu) = T(c(x1, x2)) = T(cx1, cx2) = (3(cx1) - 5(cx2), (cx1) + 2(cx2))
= (c(3x1) - c(5x2), c(x1) + c(2x2))
= c(3x1 - 5x2, x1 + 2x2)
= cT(x1, x2)
Hence, T satisfies the homogeneity condition.
Since T satisfies both the additivity and homogeneity conditions, we can conclude that the transformation T(x1,x2) = (3x1 - 5x2, x1 + 2x2) is linear.