The set of all vectors of the form
The set of all vectors of the form
(a) \(\left[\begin{array}{l}a \\ b \\ 1\end{array}\right]\)
(b) \(\left[\begin{array}{c}a \\ b \\ a+2 b\end{array}\right]\)
(c) \(\left[\begin{array}{l}a \\ 0 \\ 0\end{array}\right]\)
(d) \(\left[\begin{array}{l}a \\ b \\ c\end{array}\right]\), where \(a+2 b-c=0\)
Homework Answers
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...
Show that set of all vectors of the form (a, b, c, d) of R4...
Show that set of all vectors of the form (a, b, c, d) of R4 such that a = b + c + d is subspace of R4, whereas the set of all vectors of the form (a, b, c, d) of R4 such that a = b + c + d + 2 is not subspace of R4
minimal realizations
Problem2: Minimal Realizationsa: Find a minimal realization of the following system:$$ \begin{array}{l} \dot{x}(t)=\left[\begin{array}{cc} -1 & 1 \\ 0 & -2 \end{array}\right] x(t)+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] u(t) \\ y(t)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x(t) \end{array} $$b: Check if the following realization is minimal:$$ \dot{x}(t)=\left[\begin{array}{cc} -1 & 1 \\ 0 & -2 \end{array}\right] x(t)+\left[\begin{array}{l} 0 \\ 1 \end{array}\right] u(t) $$$$ y(t)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x(t) $$ci Consider a single-input, single-output system given by:$$ \begin{array}{l} \dot{x}(t)=\left[\begin{array}{cccc} -2 & 3 & 0...
Consider the linear system A.r = b where A = 1
Consider the linear system \(A x=b\) where \(A=\left[\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right], b=\left[\begin{array}{l}1 \\ 1\end{array}\right], x=\left[\begin{array}{l}1 \\ 1\end{array}\right]\).We showed in class, using the eigenvlaues and eigenvectors of the iteration matrix \(M_{G S}\), that for \(x^{(0)}=\left[\begin{array}{ll}0 & 0\end{array}\right]^{T}\) the error at the \(k^{t h}\) step of the Gauss-Seidel iteration is given by$$ e^{(k)}=\left(\frac{1}{4}\right)^{k}\left[\begin{array}{l} 2 \\ 1 \end{array}\right] $$for \(k \geq 1\). Following the same procedure, derive an analogous expression for the error in Jacobi's method for the same system.
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 \(]\)
roblem 1: Consider the set of all vectors in R1 which are mutually orthogonal to the vectors <3,4,-1,1> and (a) The first thing you need to do is determine the form of all vectors in this space...
roblem 1: Consider the set of all vectors in R1 which are mutually orthogonal to the vectors <3,4,-1,1> and (a) The first thing you need to do is determine the form of all vectors in this space. Hints on how to proceed You need vectors < a,b,c,d> with the property that <a,b,c,d> is orthogonal to <3,4,-1,1>and <a,b,c,d is orthogonal to <1,1,0,2>. There's a vector equation that defines "orthogonal" and this will set up two equations. .That means you have two...
Help with Linear Algebra
1. Suppose that \(T\) is the matrix transformation defined by the matrix \(A\) and \(S\) the matrix transformation defined by \(B\) where$$ A=\left[\begin{array}{rrr} 3 & -1 & 0 \\ 1 & 2 & 2 \\ -1 & 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & -1 & 0 \\ 2 & 1 & 2 \end{array}\right] $$a. If \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\), what are the values of \(m\) and \(n ?\) What values of \(m\) and \(n\) are appropriate for the...
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 \(\vec{u} \times \vec{v}\)...
Show that the following are not vector spaces: (a) The set of all vectors [x, y] in R...
Show that the following are not vector spaces: (a) The set of all vectors [x, y] in R^2 with x ≥ y, with the usual vector addition and scalar multiplication. ------------------------------------------------[a b] (b) The set of all 2×2 matrices of the form [c d] in where ad = 0, with the usual matrix addition and scalar multiplication. I need help with this question. Could you please show your work and the solution.
2. Determine which of the following are subspaces of R3: (a) all vectors of the form...
2. Determine which of the following are subspaces of R3: (a) all vectors of the form (a,b,c)), where a - 2b = c. (b) all vectors of the form (a, b, -3)), (c) all vectors of the form (a, b,0)). Explain your answer.
roblem 1: Consider the set of all vectors in R1 which are mutually orthogonal to the vectors <3,4,-1,1> and (a) The first thing you need to do is determine the form of all vectors in this space. Hints on how to proceed You need vectors < a,b,c,d> with the property that <a,b,c,d> is orthogonal to <3,4,-1,1>and <a,b,c,d is orthogonal to <1,1,0,2>. There's a vector equation that defines "orthogonal" and this will set up two equations. .That means you have two...
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 \(\vec{u} \times \vec{v}\)...
2. Determine which of the following are subspaces of R3: (a) all vectors of the form (a,b,c)), where a - 2b = c. (b) all vectors of the form (a, b, -3)), (c) all vectors of the form (a, b,0)). Explain your answer.
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