[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_VS (i.e. ordering by B_V first, and then B_S), find the matrix representation for (R T) (that is, find (R T)_BAB_VS).

Problems continue on the next page.

1

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|

|jiBA

|iiBB

(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|

|jiBA |iiBB

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_VS, find the matrix representation for (R T)

(that is, find (R T)_BVSBAB). [Hint: Kronecker product.]