MATH 235 W22 - A5
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH 235 W22 - A5
Question 1
Let V, W, and X be vector spaces.
Let T1 : V - W and T2 : W - X be linear transformations.
Consider the composite linear transformation T2 。T1 : V - X, defined by
(T2 。T1 )(v) = T2 (T1 (v)), Vv e V.
Part I
(a) Show that if both T1 and T2 are onto, then T2 。T1 is also onto.
(b) Show that if T2 。T1 is onto, then T2 is onto.
(c) Provide a counterexample to show that if T2 。T1 is onto, then T1 need not be onto. In your counterexample, clearly state the range of (T1 ), i.e. n(T1 ).
Part II
(a) Show that if both T1 and T2 are one-to-one, then T2 。T1 is also one-to-one. (b) Show that if T2 。T1 is one-to-one, then T1 is one-to-one.
(c) Provide a counterexample to show that if T2 。T1 is one-to-one, then T2 need not be one-to-one. In your counterexample, clearly state the nullspace of T2 , i.e. N(T2 ).
Question 2
Let T : V - W be a linear transformation where V and W are two vector spaces. Let X = n(T). Consider the linear transformation L : V - X, defined by
L(v) = T (v), Vv e V.
Show that L is onto.
Question 3
Let T : V - W be a linear transformation where V and W are two
finite-dimensional vector spaces.
(a) Show that if dim(V) > dim(W), then T cannot be one-to-one. (b) Show that if dim(V) < dim(W), then T cannot be onto.
Question 4
(a) Provide an example of a linear transformation T1 : P1 (R) - P1 (R) such that N(T1 ) = n(T1 ).
(b) Show that there does not exist a linear transformation T2 : P2 (R) - P2 (R) such that N(T2 ) = n(T2 ).
(c) Provide an example of a linear transformation T3 : P3 (R) - P3 (R) such that N(T3 ) = n(T3 ).
2022-02-16