Let $T:\mathbb{R^n}\rightarrow\mathbb{R}^m$ and $S:\mathbb{R^n}\rightarrow\mathbb{R}^m$ be linear. Show that $S\circ T$ is also linear. Carefully justify each step in the process.

I got this wrong and can't figure it out.

Question

Let $T:\mathbb{R^n}\rightarrow\mathbb{R}^m$ and $S:\mathbb{R^n}\rightarrow\mathbb{R}^m$ be linear. Show that $S\circ T$ is also linear. Carefully justify each step in the process.

I got this wrong and can't figure it out.

loser66 · Answer

Let u1, u2 be arbitrary vectors in R^n  and let c be a scalar, we must prove that S o T (u1+u2) = (SoT)(u1 ) + (SoT) (u2) and (SoT)(cu ) = c (SoT)(u)
SoT(u1+u2 )= S (T (u1 + U2)
                      = S (T(u1) + ST(u2)
                      = SoT (u1) + SoT(u2) (close under addition)
the part for scalar is too easy, right?

zarkon · Answer

the way you have it written  $S\circ T$ would not be defined

zarkon · Answer

double check the mappings of T and S

anonymous · Answer

You are correct, should be: $T:\mathbb{R^n\rightarrow\mathbb{R}^m}$ and $T:\mathbb{R^m\rightarrow\mathbb{R}^p}$.

anonymous · Answer

The second T should be S.

zarkon · Answer

that looks better ;)

anonymous · Answer

Thanks