Question

Let T be a linear transformation from R^r to R^s .
Determine whether or not T is one-to-one in each of the following situations:
1. r > s
2. r < s
3. r = s

A. T is not a one-to-one transformation
B. T is a one-to-one transformation
C. There is not enough information to tell

Explain reason clearly plz

Answer 1

A transformation T: V→W is one-to-one, if, different vectors in the domain have different images in the range i.e. T(u)=T(v) implies that u = v.

A transformation T:V→ W is onto if, for every vector in the range is the image of some vector in the domain.
(i) r > s.

In this case, T: R^r → R^s cannot be one-to-one as the domain will have more linearly independent vectors than the range. However, T can be onto as there are fewer linearly independent vectors in the range than in the domain.

(ii). r = s .

In this case, T can be both one-to-one and onto. For example, the identity transformation from Rⁿ to Rⁿ is both one-to-one and onto.

(iii). r < s

In this case, T can be one- to- one. For example, T: R² → R³ defined by T(x,y)=(x,y,0) is one- to- one. However, T cannot be onto as T is onto if and only if the dimension of the range of T = dimension of the co-domain..