2.12  Review Exercises

2.1. Given two vectors $[4,3]$ and $[-5,2]$, find their norms, inner product and angle between them.
2.2. Given two vectors $\mathbf{\text{x}}=[4,0]$ and $\mathbf{\text{y}}=[-5,2]$, find the projections ${\mathbf{\text{p}}}_{\mathit{yx}}$ and ${\mathbf{\text{p}}}_{\mathit{xy}}$ and the respective error vectors ${\mathbf{\text{e}}}_{\mathit{yx}}$ and ${\mathbf{\text{e}}}_{\mathit{xy}}$.
2.3. Is the matrix $\mathbf{\text{A}}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -5\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$ unitary? Design a unitary matrix $\mathbf{\text{B}}$ that is similar to $\mathbf{\text{A}}$ in the sense that its first basis (column) is a vector in the same direction as ${[1,3]}^T$.
2.4. Assume a 2-d vector space has non orthogonal basis given by $\overline{i}=[2,1]$ and $\overline{j}=[0,3]$. Find the coefficients $\alpha$ and $\beta$ that allow to represent the vector $\mathbf{\text{x}}=[4,1]$ as the linear combination $\mathbf{\text{x}}=\alpha \overline{i}+\beta \overline{j}$.
2.5. Assuming a 2-d vector space with orthonormal basis vectors $\overline{i}$ and $\overline{j}$, prove that inner products can be used to find the coefficients $\alpha$ and $\beta$ that allow to represent a vector $\mathbf{\text{x}}=[x_1,x_2]$ as the linear combination $\mathbf{\text{x}}=\alpha \overline{i}+\beta \overline{j}$.
2.6. Find the inner product between the signals: a) $u(t)$ and $e^{-0.9t}$, b) $\text{cos}<!--\; FUNCTION\; APPLICATION\; -->(0.5\mathit{\pi n})$ and $\delta [n]$, c) $u[-n-1]$ and $u[n]$.