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 prove that it satisfies the definition of linearity.
The definition of linearity requires two conditions to be met:
1. T(u + v) = T(u) + T(v) for all u, v in R^2 (additivity)
2. T(cu) = cT(u) for all u in R^2 and c in R (homogeneity)
Let's first verify the additivity condition:
1. T(u + v) = T((x1, x2) + (y1, y2))
= T((x1 + y1, x2 + y2))
= (3(x1 + y1) - 5(x2 + y2), (x1 + y1) + 2(x2 + y2))
= (3x1 + 3y1 - 5x2 - 5y2, x1 + y1 + 2x2 + 2y2)
Now let's calculate T(u) + T(v):
T(u) = T(x1, x2) = (3x1 - 5x2, x1 + 2x2)
T(v) = T(y1, y2) = (3y1 - 5y2, y1 + 2y2)
T(u) + T(v) = (3x1 - 5x2, x1 + 2x2) + (3y1 - 5y2, y1 + 2y2)
= (3x1 + 3y1 - 5x2 - 5y2, x1 + y1 + 2x2 + 2y2)
Comparing the expressions T(u + v) and T(u) + T(v), we can see that they are equal. Therefore, the additivity condition is satisfied.
Now let's move on to verifying the homogeneity condition:
2. T(cu) = T(c(x1, x2))
= T((cx1, cx2))
= (3(cx1) - 5(cx2), (cx1) + 2(cx2))
= (3cx1 - 5cx2, cx1 + 2cx2)
Now let's calculate cT(u):
cT(u) = cT(x1, x2)
= c(3x1 - 5x2, x1 + 2x2)
= (c(3x1) - c(5x2), c(x1) + c(2x2))
= (3cx1 - 5cx2, cx1 + 2cx2)
Comparing the expressions T(cu) and cT(u), we can see that they are equal. Therefore, the homogeneity condition is satisfied.
Since the transformation T: R^2 -> R^2 satisfies both the additivity and homogeneity conditions, we can conclude that it is linear.
To show that the transformation T is linear, we need to verify that it satisfies two conditions:
1. Additivity: T(u + v) = T(u) + T(v) for all vectors u and v in R^2.
2. Homogeneity: T(cu) = cT(u) for all scalar c and vector u in R^2.
Let's check each condition to determine whether T is linear:
1. Additivity:
Let's take two arbitrary vectors u = (x1, x2) and v = (y1, y2) in R^2.
T(u + v) = T((x1, x2) + (y1, y2))
= T((x1 + y1, x2 + y2))
= (3(x1 + y1) - 5(x2 + y2), (x1 + y1) + 2(x2 + y2))
= (3x1 + 3y1 - 5x2 - 5y2, x1 + y1 + 2x2 + 2y2)
= (3x1 - 5x2, x1 + 2x2) + (3y1 - 5y2, y1 + 2y2)
= T(u) + T(v)
Since T(u + v) = T(u) + T(v) holds for any vectors u and v, the additivity property is satisfied.
2. Homogeneity:
Let's take an arbitrary scalar c and a vector u = (x1, x2) in R^2.
T(cu) = T(c(x1, x2))
= T((cx1, cx2))
= (3(cx1) - 5(cx2), cx1 + 2(cx2))
= c(3x1 - 5x2, x1 + 2x2)
= cT(u)
Since T(cu) = cT(u) holds for any scalar c and vector u, the homogeneity property is satisfied.
Since both the additivity and homogeneity conditions are satisfied, we can conclude that the transformation T: R^2 -> R^2 given by T(x1, x2) = (3x1 - 5x2, x1 + 2x2) is a linear transformation.