c + a = -b
$$ \begin{aligned} &\vec{c}+\vec{a}=-\vec{b} \\ &\vec{c}-\vec{a}=-\vec{b} \\ &\vec{c}+\vec{a}=\vec{b} \\ &\vec{c}-\vec{a}=\vec{b} \end{aligned} $$
Homework Answers
Find the Trace of B
3. Let \(\quad B=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\).(a) Find the Trace of B.(b) Find \(B^{-1}\), the inverse of \(B\).(c) A vector \(\vec{v}\) is an eigenvector of the matrix \(B\) if Matrix-Vector Multiplication \(B \vec{v}\) results in a scaling of the vector \(\vec{v}\). (i.e. \(B \vec{v}=c \vec{v}\), with \(c\) a real number.) Using the definition of Matrix-Vector Multiplication show that the vector \(\vec{v}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) is an eigenvector of \(B\) with eigenvalue \(c=3\).
Given an LTI system with
Given an LTI system with$$ \begin{aligned} &A=\left(\begin{array}{cc} 1 / 2 & 0 \\ 0 & -1 / 4 \end{array}\right), B=\left(\begin{array}{l} 0 \\ 1 \end{array}\right), C=(1-1), \\ &D=0 \quad X(0)=\left(\begin{array}{l} -1 \\ -1 \end{array}\right), U(n)=(-1)^{n} u[n] \end{aligned} $$Calculate \(y[n], y[4]\) and \(y[\) Steady State \(]\)
C++ int num [ ] = { 1,2,3,3,4,5,6,7,8 } vector <int> vec (num, num + 8)...
C++ int num [ ] = { 1,2,3,3,4,5,6,7,8 } vector <int> vec (num, num + 8) ; vector <int> : : iterator iter ; iter = vec . begin ( ) ; Question 1. What is the result of execute the instruction num [4] ;? a)2 b)7 c) unknown Question 2. How many elements does the vector vec contain? a)0 b)10 c)16
Find the general solution for
Q1) Find the general solution for \(\vec{x}^{\prime}=\left[\begin{array}{cc}2 & 1 \\ -3 & 6\end{array}\right] \vec{x}\).Q2) Find the general solution for \(\vec{x}^{\prime}=\left[\begin{array}{ll}-1 & 1 \\ -4 & 3\end{array}\right] \vec{x}\).
Defining the cross product The cross product of two nonzero vectors
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
1. Solve the system: x' =3x+5y, y' =-x-y2. Find the general solution to
Solve the system: \(x^{\prime}=3 x+5 y, y^{\prime}=-x-y\)Find the general solution to$$ \vec{x}^{\prime}=\left(\begin{array}{ll} 2 & 1 \\ 0 & 2 \end{array}\right) \vec{x} $$Find the general solution to$$ \vec{x}^{\prime}=\left(\begin{array}{ccc} 3 & 0 & -2 \\ 0 & 5 & 0 \\ 2 & 0 & 3 \end{array}\right) \vec{x} $$
Referring to the vectors in the figure, express the sum A + B +C in unit vector notation
Referring to the vectors in the figure, express the sum {rm vec A} + {rm vec B} + {rm vec C} in unit vector notation
I have the first method complete, but I can't figure out thesecond method. I have...
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
linear algebra
Problem settingConsider the linear transformation \(\phi(\cdot): \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) on the standard vector space of dimension two over the field of real numbers defined as:$$ \phi\left(\left(\begin{array}{l} x_{0} \\ x_{1} \end{array}\right)\right)=\left(\begin{array}{r} 3 x_{0}-x_{1} \\ -7 x_{0}+2 x_{1} \end{array}\right) $$Problem taskFind \(\mathcal{R}_{G \rightarrow E}(\) id \()\) that is the change of basis matrix from basis \(G\) to the standard basis \(E\) where the standard basis vectors are:$$ \begin{array}{l} \vec{e}_{0}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right) \\ \vec{e}_{1}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) \end{array} $$given that...
1. A triangular pulse (A) moving from left to right passes through a triangular pulse (B)...
1. A triangular pulse (A) moving from left to right passes through a triangular pulse (B) moving from right to left. Using graph paper, draw the superposition of these pulses when (a) they begin to pass through each other, (b) when they are aligned, and (c) as they are close to completely passing through each other.
3. Let \(\quad B=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\).(a) Find the Trace of B.(b) Find \(B^{-1}\), the inverse of \(B\).(c) A vector \(\vec{v}\) is an eigenvector of the matrix \(B\) if Matrix-Vector Multiplication \(B \vec{v}\) results in a scaling of the vector \(\vec{v}\). (i.e. \(B \vec{v}=c \vec{v}\), with \(c\) a real number.) Using the definition of Matrix-Vector Multiplication show that the vector \(\vec{v}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) is an eigenvector of \(B\) with eigenvalue \(c=3\).
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
1. A triangular pulse (A) moving from left to right passes through a triangular pulse (B) moving from right to left. Using graph paper, draw the superposition of these pulses when (a) they begin to pass through each other, (b) when they are aligned, and (c) as they are close to completely passing through each other.
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