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For differential equation x''+0.4x'+1.69x=cos t , with initial conditions x(0)=0 and x'(0)=0, use a numerecal solver to plot solution. Find the time constant and find intelval where the transient response has died out. On this interval draw the graph of forcing function cos t and find the gain and phase.

Mark each statement True or False. Justify each answer. a. If $\left\{\mathbf{v}_{1}, \dots, \mathbf{v}_{p}\right\}$ is an affinely dependent set in $\mathbb{R}^n$ , then the set $\left\{\tilde{\mathbf{v}}_{1}, \ldots, \tilde{\mathbf{v}}_{p}\right\}$ in $\mathbb{R}^{n+1}$ of homogeneous forms may be linearly independent. b. If $v_1, v_2, v_3$ , and $v_4$ are in $\mathbb{R}^3$ and if the set $\left\{\mathbf{v}_{2}-\mathbf{v}_{1}, \mathbf{v}_{3}-\mathbf{v}_{1}, \mathbf{v}_{4}-\mathbf{v}_{1}\right\}$ is linearly independent, then $\left\{\mathbf{v}_{1}, \dots, \mathbf{v}_{4}\right\}$ is affinely independent. c. Given $S=\left\{\mathbf{b}_{1}, \dots, \mathbf{b}_{k}\right\}$ in $\mathbb{R}^n$ , each p in aff S has a unique representation as an affine combination of $\mathbf{b}_{1}, \dots, \mathbf{b}_{k}$ . d. When color information is specified at each vertex $v_1, v_2, v_3$ of a triangle in $\mathbb{R}^3$ , then the color may be interpolated at a point p in $\operatorname{aff}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ using the barycentric coordinates of p. e. If T is a triangle in $\mathbb{R}^2$ and if a point p is on an edge of the triangle, then the barycentric coordinates of p (for this triangle) are not all positive.