# Linear Transformation Composition Proof

• Jul 7th 2012, 01:22 PM
PiIsCool
Linear Transformation Composition Proof
I need help with this linear algebra proof.

Show that the composition of a linear transformation is also a linear transformation. Let T: Rn to Rm and S: Rm to Rp be linear transformations. Show that S(T): Rn to Rp is also a linear transformation. Show that the composition respects scalar multiplication and vector addition.

Thanks.
• Jul 7th 2012, 03:57 PM
ModusPonens
Re: Linear Transformation Composition Proof
S(T(u+v))=S(T(u)+T(v))=S(T(u))+S(T(v))

Now you can do the scalar multiplication.
• Jul 8th 2012, 09:20 AM
Deveno
Re: Linear Transformation Composition Proof
note that the composition is usually written ST, or SoT, not S(T) (S doesn't have "linear transformations" as its input, but the VECTORS T(v)).
• Jul 8th 2012, 01:23 PM
PiIsCool
Re: Linear Transformation Composition Proof
II. Show that the composition of a linear transformation is also a linear transformation. Let T: Rn Rm and S: Rm Rp be linear transformations. Show that (S o T)(x) = S(T(x)). Show that the composition respects scalar multiplication and vector addition.
• Jul 8th 2012, 03:22 PM
Deveno
Re: Linear Transformation Composition Proof
(SoT)(x) is by DEFINITION the function that takes x to S(T(x)):

x-->T(x)-->S(T(x))

so there's nothing to show. what there is to show is that:

a)(SoT)(x+y) = (SoT)(x) + (SoT)(y) for all vectors x and y
b) (SoT)(ax) = a(SoT(x)), for all scalars a, and all vectors x.

ModusPonens already proved (a) in post #2, can you prove (b)?

this is mostly "manipulating the definitions"...it sounds as if you are unsure of what it is you are being asked to prove. so, how do we know a function from one vector space to another is a linear transformation?