Linear Transformation Proof?

Does anyone know how to prove this?

Recall composition of functions. We write g(f(x)) if we perform f then perform g, and if the target space of f is in the domain of g. For sets of vectors X, Y,Z let T1 : X -%26gt; Y and T2 : Y -%26gt; Z be linear transformations. Show that the composition T2(T1(a)) is a linear transformation. Here you鈥檒l need to use the two linear conditions of T1 and T2 to show the composition is linear.

Linear Transformation Proof?ballet

T1 : X --%26gt; Y and T2 : Y --%26gt; Z

T1(a) is a mapping from X to Y.Since T1 is linear there exists a value b for the mapping T1(a).

Now T2(b) is mapping from Y to Z and T2 being linear we can say for some c ....T2(b) = c

So net we have T2(T1(a)) = c

Thus we can conclude that the above mapping is linear.

Linear Transformation Proof?globe theater opera theater

Suppose a and b are scalars and u and v are vectors. Now

T2(T1(au + bv)) = T2(T1(au) + T1(bv)) = T2(aT1(u) + bT1(v)) =

T2(aT1(u)) + T2(bT1(v)) = aT2(T1(u)) + bT2(T1(v)). Therefore T2(T1)) is linear.