Math 117 - Summer 2022 Homework 3
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Homework 3
Math 117 - Summer 2022
1) (4 points) Let V be finite dimensional and let W ⊆ V be a subspace. Recall the defi-
nition of the annihilator of W, Wo from class. Prove using dual basis that dim(Wo ) = dim(V ) − dim(W)
(hint: extend basis....)
2) (3 points) Let V be any vector space (potentially infinite dimensional). Prove that (V\W)∗ ≃ Wo
(Hint: Universal property of quotient....)
Remark: This isomorphism gives another proof of problem 1, in the case when V is finite dimensional
3) (3 points) Let V, W be finite dimensional vector spaces over F and let T ∶ V → W be a linear map. Recall the isomorphism we constructed in class ΦV ∶ V → V ∗∗ by sending v to evv . Prove that the following diagram commutes
ΦV
∗∗
∗∗
ie, that ΦW ○ T = T ∗∗ ○ ΦV (Hint: Recall that T ∗∗ ∶ V ∗∗ → W ∗∗ sends a linear functional φ ∶ V ∗ → F to the linear functional φ○ T∗ ∶ W ∗ → F. That is T∗∗ (φ) = φ○ T∗ ∈ W ∗∗ . You will then evaluate what this is on a linear functional γ ∈ W ∗ )
4) Let V be an n-dimensional vector space. We call a subspace of dimension n-1 a hyperplane. (a) (1 point) If φ ∶ V → F is a nonzero linear functional, prove that ker(φ) is a hyperplane
(b) (2 points) Prove moreover that every hyperplane is the kernal of a nonzero linear
functional.
(c) (2 points) More generally, prove that a subspace of dimension d is the intersection of n-d hyperplanes (ie, from part b, is the intersection of n-d kernals of linear functionals). (Hint: Dual basis can be helpful here...)
5) Let V, W be finite dimensional vector spaces over F.
(a) (3 points) Prove that
V ⊗ W ∗ ≃ L(V,W)
(b) (2 points) Use this to prove that
(V ⊗ W)∗ ≃ L(V,W ∗ )
Unimportant Remark: Writing out the duals, this isomorphism above is saying that L(V ⊗ W,F) ≃ L(V,L(W,F))
That is, maps out of the tensor product of V and W into F correspond to maps from V into maps from W to F. Such a result is in fact true more generally if we replace F with any other vector space, and is a foundational result in category theory/algebra: the so called “tensor-hom adjunction.” Cool stuff
6) Consider the following vector spaces with corresponding basis:
V1 =R3
W1 =R[t]≤2
V2 =M2×2 (R)
W2 =M2×2 (R)
BV1 = {e1 ,e2 ,e3 }
BW1 = {1,t,t2 }
1 0 0 1 0 0 0 0
BV2 = {(0 0) , (0 0) , ( 1 0) , (0 1)}
1 0 0 1 0 0 0 0
BW2 = {(0 0) , (0 0) , ( 1 0) , (0 1)}
Now consider the following two linear transformations T1 ∶ V1 → W1 and T2 ∶ V2 → W2 given by
⎛a ⎞
T1 (⎜b ⎟) = a + b − ct + at2
⎝ c⎠
⎛ a1 a2 ⎞ 2a2 a4
(a) (1 point) Write the corresponding basis for V1 ⊗ V2 and W1 ⊗ W2
(b) (4 points) Recall we get the linear map
T1 ⊗ T2 ∶ V1 ⊗ V2 → W1 ⊗ W2
(T1 ⊗ T2 )(v1 ⊗ v2 ) = T1 (v1 ) ⊗ T2 (v2 )
Compute the matrix of this map with respect to the two basis you found in part a
Unimportant Rmk: This is an example of what is called the Kronecker-Product of ma- trices. It is an operation that takes an m × n and a k × l matrix and produces an mk × nl matrix. This matrix is precisely the matrix of the tensor product of linear maps we defined in class/on your mini-hw. I recommend looking it up- its a pretty cool thing.
2022-07-15