Question

(1) Let T: Rn--->Rm be linear tranformations.

a. If T maps Rnonto Rm, give a relationship between m and n

b. If T is one-to-one, give a relationship between m and n

c. If T maps Rn onto Rm and is one-to-one, give a relationship between m and n

(Hint: Think about the size of the standard matrix representation of T and the placement of the pivots in each case)

(2) Let T: R3 ---> R4 be a linear transformation such that the only solution to T(x) = 0 is trivial solution.

a. If T is one-to-one

b. Does T map R3onto R4?

Justify your answers in each case.

(Hint:one way to approach this is to look at what the martix representation of T might look like and where it does or does not have pivots.)

(3) Suppose a linear transformation T: R2----> R2 is formed by taking a rotation counterclockwise of 90 degrees, follwed by a reflection through the X2-axis. Describe the points that will be moved back to their original position by this transformation?

(Hint: Think about what T will do to the unit box and the vectors e1 and e2)

Answer 1

Check the explanation

Explanation:

1.

(a)

n>=m

(b)

n <= m

(c)

n=m

2.

(a)

let T(v1) = T(v2)

=>

T(v1)-T(v2) = 0

=>

T(v1-v2) = 0

=>

v1-v2 = 0 from the hypothesis

=>

v1=v2

=>

T is one-one

thus proved

(b)

lets assume T is onto, we already know that T is one-one, so from above problem (third case where m=n)

we should have 3=4 which is impossible

so T is NOT onto.

3.

we need to find a,b such that

T(a,b) =(a,b)

=>

a= b

=>

points on the line x=y are the required points