Question

Suppose T : X Y and U:Y → Z are linear maps. Without referring to matrices, show that UOT: X Z is linear. Note where you use that T is linear and that U is linear.

Answer 1

To show that UOT: X → Z is linear, we need to prove that it satisfies the properties of linearity. Since U: Y → Z and T: X → Y are linear maps, we can use the properties of linearity to show that UOT is also linear.

Step-by-step explanation:

To show that UOT: X → Z is linear, we need to prove that it satisfies the properties of linearity. Since U: Y → Z and T: X → Y are linear maps, we can use the properties of linearity to show that UOT is also linear.

Let's consider two vectors x1 and x2 in X and a scalar α.

1. Linearity with respect to addition:

UOT(x1 + x2) = U(OT(x1 + x2)) = U(O(T(x1) + T(x2))) = U((OT(x1) + OT(x2))) = UOT(x1) + UOT(x2)

2. Linearity with respect to scalar multiplication:

UOT(αx1) = U(OT(αx1)) = U(O(αT(x1))) = αU(O(T(x1))) = αUOT(x1)

Thus, we can conclude that UOT: X → Z is linear.