(1) Let T : R n ---> R m be linear tranformations. a. If T maps R n onto R m , g
ID: 2966821 • Letter: #
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)
Explanation / Answer
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
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