### Homework 0

Important: This is an individual assignment.

The total for this homework is (0 points) and no submission is required. It is designed to check whether you are comfortable with the linear algebra and mathematical concepts that will be used in this class. If you have trouble with many or most of these questions, you will need to spend some time learning/reviewing. Do not turn this homework in. Check your answers with the solution.

1. #### Complex Numbers

1. What is the complex conjugate $$z^*$$ or $$\bar{z}$$ of $$z = a + ib$$?

2. What is the sum of $$a + ib$$ and $$c + id$$?

3. What is the product of $$a + ib$$ and $$c + id$$?

4. What is the magnitude $$|z|$$ of a complex number $$z?$$

5. What is an alternate representation of $$e^{i\theta}$$?

6. What is the magnitude of $$e^{i\theta}$$?

7. What are the values of $$i^2$$, $$i^3$$, and $$i^4$$?

8. For a complex value $$z$$, what is the value of $$zz^*$$?

9. What are the values of $$e^{i\frac{\pi}{2}}$$, $$e^{i\pi}$$, $$e^{i\frac{3\pi}{2}}$$, and $$e^{i2\pi}$$?

10. A number $$\omega$$ is an "$$n$$-th root of unity" for some integer $$n$$, if $$\omega^n = 1$$. For a given integer $$n$$, there are $$n$$ such roots. What is a general expression to find the $$n$$-th roots of unity? (Hint: Use Euler's notation, as in Question 5.)

2. #### Linear Algebra

1. What is the mathematical notation for a column vector with three values: $$1.2, 3.5, -0.05$$? (By default, when we say "vector," we mean "column vector.")

2. Given the vectors a and b below, what is the dot product (or inner product) a . b?
\begin{equation*} \mathbf{a} = \left[\begin{matrix}1\\2\\3\\\end{matrix}\right],\ \mathbf{b} = \left[\begin{matrix}4\\5\\6\\\end{matrix}\right] \end{equation*}
3. What is the transpose $$\mathbf{x}^\top$$ of the vector $$\mathbf{x} = \left[\begin{matrix}1\\2\\3\\\end{matrix}\right]$$ ?

4. What is the conjugate transpose $$\mathbf{x}^\dagger$$ of vector $$\mathbf{x} = \left[\begin{matrix}a\\b\\c\\\end{matrix}\right]$$, where $$a, b,$$ and $$c$$ are complex numbers?

5. Given the complex vector $$\mathbf{x}$$ from the previous question, what is $$\mathbf{x}^\dagger \mathbf{x}$$?

6. Compute the product: $$\left[\begin{matrix}a\\b\\c\\\end{matrix}\right]\ \left[\begin{matrix}x&y&z\\\end{matrix}\right]$$.

7. Compute the matrix-vector product $$\left[\begin{matrix}a&b\\c&d\\\end{matrix}\right]\ \left[\begin{matrix}x\\y\\\end{matrix}\right]$$.

8. Compute the vector-matrix product $$\left[\begin{matrix}a&b&c\\\end{matrix}\right]\ \left[\begin{matrix}x_0&x_1&x_2\\y_0&y_1&y_2\\z_0&z_1&z_2\\\end{matrix}\right]$$.

9. Compute the matrix product $$\left[\begin{matrix}a&b&c\\d&e&f\\h&j&k\\\end{matrix}\right] \left[\begin{matrix}r&s&t\\u&v&w\\x&y&z\\\end{matrix}\right]$$.

10. What is the representation of a 4-dimensional identity matrix?

11. For a square matrix $$A$$, what is the relationship between $$A$$ and its inverse $$A^{-1}$$?

12. In general, for square matrices $$A$$ and $$B$$, is it true that $$AB = BA$$?

13. If $$A\mathbf{x} = \lambda\mathbf{x}$$, where $$A$$ is a square matrix, $$\mathbf{x}$$ is a vector, and $$\lambda$$ is a scalar, what is the value $$\lambda$$ called?

14. If $$A\mathbf{x} = \lambda\mathbf{x}$$, where $$A$$ is a square matrix, $$\mathbf{x}$$ is a vector, and $$\lambda$$ is a scalar, what is the value $$\mathbf{x}$$ called?

15. What is the transpose $$A^\top$$ of the matrix $$A\left[\begin{matrix}a&b\\c&d\\\end{matrix}\right]$$?

16. What is the conjugate transpose $$A^\dagger$$ of the matrix $$A\left[\begin{matrix}a&b\\c&d\\\end{matrix}\right]$$?

17. A square matrix with exactly one 1 in each row and column is called a permutation matrix. Compute the result of this matrix-vector multiplication: $$\left[\begin{matrix}0&1&0&0\\0&0&0&1\\0&0&1&0\\1&0&0&0\\\end{matrix}\right] \left[\begin{matrix}a\\b\\c\\d\\\end{matrix}\right]$$.

18. A unitary matrix is one for which $$U^{\dagger}U = UU^\dagger = I$$. Show that each of the matrices below is unitary.
\begin{equation*} X = \left[\begin{matrix}0&1\\1&0\\\end{matrix}\right],\ Y = \left[\begin{matrix}0&-i\\i&0\\\end{matrix}\right],\ Z = \left[\begin{matrix}1&0\\0&-1\\\end{matrix}\right],\ H = \frac{1}{\sqrt{2}}\left[\begin{matrix}1&1\\1&-1\\\end{matrix}\right] \end{equation*}

19. What is the trace of the matrix $$A = \left[\begin{matrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{matrix}\right]$$, written as tr ($$A$$)?

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