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Problem Set 2
Math 423, Section 200, Spring 2015
Due: Friday, February 6.
Review Chapter 3 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Friday, February 6. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1. Suppose that T ∈ L (V, W) and v1 , . . . , vm ∈ V such that T v1 , . . . , T vm are linearly independent.
Prove that v1 , . . . , vm are linearly independent.
2.
a. Give an example of a function ψ : R2 → R such that
ψ(λv) = λψ(v)
for all λ ∈ R and v ∈ R2 , but ψ is not linear.
b. Give an example of a function ϕ : C → C such that
ϕ(w + z) = ϕ(w) + ϕ(z)
for all w, z ∈ C, but ϕ is not linear. (Here C is thought of as a C-vector space.)
3. Show that {T ∈ L (R5 , R4 ) | dim ker T > 2} is not a subspace of L (R5 , R4 ).
4.
a. Give an example of a map T : R4 → R4 such that range T = ker T .
b. Show that there does not exist a map T : R5 → R5 such that range T = ker T .
5. Suppose that V is finite dimensional with dim V > 0, and W is infinite dimensional. Show that
L (V, W) is infinite-dimensional.
6. Suppose that V is finite dimensional, and T ∈ L (V). Prove that T is a scalar multiple of the
identity if and only if S T = T S for all S ∈ L (V).
7.
a. Find linear transformations S , T : R2 → R2 such that S T = 0 but T S , 0.
b. Find a nonzero linear transformation T : R2 → R2 such that T T = 0.
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8. Note that C is an R-vector space (we can add complex numbers, and multiply them by real
numbers).
a. Show that C is isomorphic to R2 as an R-vector space.
√
b. Let a, b ∈ R, and consider λ = a + b −1 ∈ C. Show that multiplication by λ gives a
linear transformation from the R-vector space C to the R-vector space C.
c. Find a matrix for the transformation in the previous part.
9. Show that V × V and L (F2 , V) are isomorphic vector spaces.
10. Suppose that T is a function from V to W, where V and W are F-vector spaces. Define the
graph of T to be the set
{(v, T v) | v ∈ V} ⊆ V × W.
Show that T is a linear map if and ony if the graph of T is a subspace of V × W.
11. Suppose that V, W are vector spaces such that V × W is finite dimensional. Show that V and W
are finite dimensional.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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