(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)

Check the explanation

Step-by-step 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

pay attention in your classes and concentrate the best thing you can do to raise your test scores is to pay attention when you're supposed to be learning the material: in class!

take good notes.

do your homework.

use mnemonics and other tricks.

do practice tests.

note: the best i know.

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