Let T1 Rm rightarrow Rk and T2 Rk rightarrow Rn If T1 is one

Let T_1: R^m rightarrow R^k and T_2: R^k rightarrow R^n. If T_1 is one-to-one, what can you say about m and k? If T_2 T_2 is one-to-one, prove that T_1 is one-to-one.

Solution

A)

One to one map means no two elements have the same image.

Dimension of R^m is m so R^k must have a dimension at least m ie k>=m

B)

Let,
T_1 not be one to one

So there exists x ,y so that x not equal to y such that:T_1(x)=T_1(y)

Hence, T_2(T_1(x))=T_2(T_1(y))

(T_2oT_1)(x)=(T_2oT_1)(y)

But, T_2oT_1 is one to one so ,x=y. This is a contradiction.

Hence, T_1 is one to one.