True or false:
$$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x \geq 0\right\} $$
is a subspace of R3.
True
False
Question 10 (1 point)
True or false:
$$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x-y=z+1\right\} $$
is a subspace of R3.
True
False
Problem on Linear programming and Simplex methodThe \(\ell_{1}\) norm of a vector \(v \in \mathbb{R}\) is defined by$$ \|v\|_{1}:=\sum_{i=1}^{n}\left|v_{i}\right| $$Problems of the form Minimize \(\|v\|_{1}\) subject to \(v \in \mathbb{R}^{n}\) and \(A v=b\) arise very frequently in applied math, particularly in the field of compressed sensing.Consider the special case of this problem whith \(n=3\),$$ A=\left(\begin{array}{lll} 1 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right) \quad \text { and } \quad b=\left(\begin{array}{l} 3 \\ 8 \end{array}\right) $$(a) (3...
Suppose that the functions \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}, g: \mathbb{R}^{3} \rightarrow \mathbb{R}\), and \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}\) are continuously differentiable and let \(\left(x_{0}, y_{0}, z_{0}\right)\) be a point in \(\mathbb{R}^{3}\) at which$$ f\left(x_{0}, y_{0}, z_{0}\right)=g\left(x_{0}, y_{0}, z_{0}\right)=h\left(x_{0}, y_{0}, z_{0}\right)=0 $$and$$ \left\langle\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right) \times \nabla h\left(x_{0}, y_{0}, z_{0}\right)\right\rangle \neq 0 $$By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a...
3. For each of the following sets, determine if it is a subspace of R3. If it is a subspace, prove it. If is is not a subspace provide an example showing how it violates at least one of the subspace axioms (a) B , y,z) E R3 (x, y, 2)l 1) (b) S (a b, 3b+ 2a,a-b) a, be R) [10 (c) P (7,5,8) s(1,-1,2)t (3, 1,4) s,te R)
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...
Q 1 Let V C R3 be the subspace V = {(x,y, z) E R3 : 5x 2y z 0} a) Find a basis B for V. What is the dimension of V? b) Find a basis B' for R3 so that B C B'
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
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...
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.
True or False 1.aThe orthogonal complement of the subspace V = {x ∈ R^3 | x1 = x3} is a line. b. If A has QR factorization A = QR, then the columns of R span Col(A) c. The least squares solution, xˆ, to a linear system Ax = b is the vector that minimizes the quantity ||Axˆ −b||^2 . d. If A ∈ R n×n and x and y are vectors in R^n , then Ax · Ay =...
Question 1 (10 Marks) This question consists of 10 true false ansers. In cach ease, answer true if the statement is always true and false otherise. If a statement is false, 1. The set rER0 isa group under the binary operation o defined ad-be is a group under matrix addition. 3. Tho sot eRzs not an Abelian group under the binary erplain why. There is no need to show working for true statements. by a ob vab. 2. The set...