[SOLVED] Quantum Homework 5

1. Let V and S be vector spaces over C with bases B_V and B_S, respectively. Define

and

and recognize it as a vector space by coordinate-wise interpretation of the vector space axioms. That is,

If R : A → V and T : A → S are linear functions, then we can define a linear function (R × T) : A → V × S by

for a ∈ A.

• Let

and .

Show that C-span(C) = V × S but that C is not always a basis for V × S.

• Prove that

and

is a basis for V × S. What is the dimension of V × S?

• Let R : A → V and T : A → S be linear functions. Suppose that A, V, and S have ordered bases

,          ,              B_S = {s₁,s₂},

and that the matrix representations of R and T relative to these bases are

and         ( .

Using the lexicographic order for the basis B_V×_S (i.e. ordering by B_V first, and then B_S), find the matrix representation for (R × T) (that is, find (R × T)_BA→B_V×_S).

Problems continue on the next page.

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                                                  QUANTUM ALGORITHMS, HW 5 ADDITIONAL PROBLEMS                                                                         2

2. Let T : A → B be a linear transformation between vector spaces with ordered bases

.

Suppose that T has matrix with respect to these bases

 .

• Show that the matrix for T can be written T = ^X a_ij |iihj|

|ji∈BA

|ii∈BB

(note that |1i ∈ B_A is a 3-dimensional vector, while |1i ∈ B_B is a 2-dimensional vector).

• Show that for fixed |ii ∈ B_A and |ji ∈ B_B

for all hv| ∈ A. From this, prove that |jihi| defines a linear transformation from A → B.

• Suppose that

R = ^X b_ij |iihj|

|ji∈BA |ii∈BB

for b_ij ∈ C. Use the previous part to prove that R is a linear transformation from A → B.

3. Let V and S be vector spaces over C with bases B_V and B_S, respectively.
   • Prove that

and

is a basis of V ⊗ S. What is the dimension of V ⊗ S?

• Let R : V → A and T : S → B be linear functions. Suppose that A, V, and S have ordered bases the same as in the previous question and that B has ordered basis and that the matrix representations of R and T relative to these bases are

and         ( .

Using the lexicographic order for the basis B_V⊗_S, find the matrix representation for (R ⊗ T)

(that is, find (R × T)_BV⊗_S→_BA⊗_B). [Hint: Kronecker product.]