Question

Let T : Rn -> Rm be a onto linear transformation. Select all the true statements from the following list.

(Incorrect choices will earn you negative points)

For any y in Rm there exists a unique x in Rn such that T(x) = y

For any y in Rm there is at most one x in Rn such that T(x) = y

The range of T is the entire Rm

For any y in Rm there is at least one x in Rn such that T(x) = y

T2, which is T composed with itself (i.e., T2(x) = T(T(x)) ) is also an onto linear transformation.

T has an inverse.

T must be one-to-one

The equation T(x) = y in x (i.e.,

Answer 1

1. For any y in R^m, there exists a unique x in Rⁿ such that T(x) = y. False as T is not stated to be one-to-one.                                               
2. For any y in R^m there is at most one x in Rⁿ such that T(x) = y. False as T is not stated to be one-to-one.                  
3. The range of T is the entire R^m. True, as T is onto.             
4. For any y in R^m there is at least one x in Rn such that T(x) = y. True, as T is onto.
5. T², which is T composed with itself (i.e., T²(x) = T(T(x)) ) is also an onto linear transformation. False. T² is not even defined as Rⁿ? R^m.                          
6. T has an inverse. False, as T is not stated to be one-to-one.        
7. T must be one-to-one. False as not every onto linear transformation is one-to-one.                       
8. The equation T(x) = y in x (i.e., INCOMPLETE