(b) From the spectral theorem, find a vector \(g(x)\) in a subspace spanned by only \(\{1, \cos(x), \cos(2x), \sin(x), \sin(2x)\}\) so that \(g(x)\) has the shortest distance to the function \(f(x) = x\) in the interval \([-\pi, \pi]\).
Note: In this problem, you may need: \(\int x \sin ax dx = -\frac{x}{a} \cos ax + \frac{1}{a^2} \sin ax\) and \(\int x \cos ax dx = \frac{x}{a} \sin ax + \frac{1}{a^2} \cos ax\)

(a) Show that the input \(n\) signals are linearly independent.

(b) Please verify that if this mixer performs a linear transformation in this system.

(a) If the period of a set of periodic functions is \(2\pi\), what is the orthonormal basis of the vector space that these periodic functions belong to? You have to show the vectors in the basis are orthonormal to get full credit.

(a) The set \(S \triangleq \{ \mathbf{A} \mid \mathbf{A} \in \mathbb{R}^{n \times n}, \mathbf{A}^T = -\mathbf{A} \}\) is not a subspace.

(b) Let \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times n}\). Suppose the system \(\mathbf{A}\mathbf{x} = \mathbf{0}_n\) has infinitely many solutions and \(\mathbf{B}\mathbf{x} = \mathbf{0}_n\) has only one solution. This implies the system \(\mathbf{A}\mathbf{B}\mathbf{x} = \mathbf{0}_n\) has exactly one solution.

(c) If \(\mathbf{A}\) has an eigenvector \(\mathbf{x}\) with eigenvalue \(\lambda\), then \(\exp(\mathbf{A})\) has \(\mathbf{x}\) as an eigenvector with the eigenvalue \(\exp(\lambda)\).

(d) If \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times n}\) and \(\mathbf{A}\mathbf{B} - \mathbf{B}\mathbf{A} = \mathbf{A}\), then \(\mathbf{A}\) is singular.

(e) Consider the following linear equations characterized by \(\mathbf{A} \in \mathbb{R}^{m \times n}\):\[\mathbf{A}\mathbf{x} = \mathbf{b}.\]Then \(\widehat{\mathbf{x}} = \mathbf{A}^{\dagger}\mathbf{b} + \mathbf{v}\) is a solution to the above linear equations, where \(\mathbf{A}^{\dagger}\) is the pseudo-inverse of \(\mathbf{A}\) and \(\mathbf{v} \in \mathcal{N}(\mathbf{A})\) (null space of \(\mathbf{A}\)). If your answer is "False", please describe the physical meaning of \(\mathbf{A}\widehat{\mathbf{x}}\).

(a) Consider a new binary random variable \(A\) which is the binary sum of \(a_1\) and \(a_2\), i.e., \(A = a_1 \oplus a_2\). Please show that the log-likelihood ratio of \(A\) defined as , can be expressed as .

(b) Please show that the probability that \(\mathbf{a}\) contains an even number of 1s is \(\frac{1}{2} + \frac{1}{2} \prod_{\ell=0}^{d-1} (1 - 2p_1^{(\ell)})\).

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