Bu = ƒ and Cu = f might be solvable even though B and C are singular. Show that every vector f = Bu has ƒ1 + ƒ2+ ……. +fn = 0. Physical meaning: the external forces balance. Linear algebra meaning: Bu = ƒ is solvable when ƒ is perpendicular to the all – ones column vector e = (1, 1, 1, 1…) = ones (n, 1).
Q: Q4: A / Given a vector v = [0:0.04:1] plot the surface of y' * y matrix when y = sin(v * 2 * )?
A: >> v = [0:0.04:1] >> y = sin(v*2*pi) >> surf(y' * y)
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A:
- Bu = ƒ and Cu = f might be solvable even though B and C are singular. Show that every
vector f = Bu has ƒ1 + ƒ2+ ……. +fn = 0. Physical meaning: the external forces balance. Linear algebra meaning: Bu = ƒ is solvable when ƒ is perpendicular to the all – ones column vector e = (1, 1, 1, 1…) = ones (n, 1).
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- (20pts) Consider the matrixA =−2 11−10 5(a) Determine, by hand, an SVD of A, A = UΣVT. The SVD is not unique,so find the one with the minimum number of minus signs in U and V . List thesingular values σ1 and σ2, and the left and right singular vectors.(b) Find A−1 not directly, but via the SVD. Check your result by typing inv(A)in MATLAB or calculating A−1 by other methods.(c) Find the eigenvalues λ1 and λ2 of A. Verify that detA = λ1λ2 and |detA| =σ1σ2Consider a vector ('L', 'M', 'M', 'L', 'H', 'M', 'H', 'H'), where H stands for ‘high’, L stands for ‘low’, and M for ‘medium’ in rstudio. a. Convert this vector into an unordered factor. b. Convert this vector into an ordered factor with Low < Medium < High.Given vectors :u = (5,2) ; v = (-2,5) w = (0,3) ; q = (10,4)4.1 Calculate the following dot products:u.v ; (u.v).w ; u.(3w) , u.(w-v)4.2 Calculate‖?‖ ; d(u,v) ; ‖? − ?‖24.3 u and v are they orthogonal4.4 u and w are they orthogonal4.5 find c real number that satisfy q = c.u4.6 Deduce that q and u are parallel4.7 Normalize vector w
- Represent the images of “space invaders” in Figure 5.16 by quantitative vectors and compute the distances between them (i.e. their similarity) using Manhattan distance measures.solve it with Julia programming 1: Modify your models and coding of Chebyshev center and building a house to get the right anwsers. # Given matrix A and vector bA = [2 -1 2; -1 2 4; 1 2 -2; -1 0 0; 0 -1 0; 0 0 -1]b = [2; 16; 8; 0; 0; 0] answer wil be:x is: 0.75y is: 3.25z is: 0.7500000000000001r is: 0.7500000000000001objective value is: 0.7500000000000001a. Build an adjacency matrix ? for this map. b. How many paths of length 2 from V5 to V1 exist? c. How many paths of length 3 from V5 to V1 exist?
- Given two squares on a two-dimensional plane, find a line that would cut these twosquares in half. Assume that the top and the bottom sides of the square run parallel to the x-axiswe represent the finite-length signals as vectors in Euclidean space, many operations on signals can be encoded as a matrix-vector multiplication. Consider for example a circular shift in C: a delay by one (i.e. a right shift) transforms the signal x = (xo X1 X2]" into x = [xı xo xz]" and it can be described by the matrix TO D = [0 1 0'0 0 1'1 0 0] so that x = Dx. Determine the matrix F that implements the one step difference operator in C ie the operator that transforms a signal x into [(x - x)(x1 - x0)(x2 - 1)]jupyter nootebook python with julia 1: Modify your models and coding of Chebyshev center and building a house to get the right anwsers. # Given matrix A and vector b A = [2 -1 2; -1 2 4; 1 2 -2; -1 0 0; 0 -1 0; 0 0 -1]b = [2; 16; 8; 0; 0; 0]
- Modify the Chebyshev center coding with julia in a simple style using vectors, matrices and for loops # Given matrix A and vector bA = [2 -1 2; -1 2 4; 1 2 -2; -1 0 0; 0 -1 0; 0 0 -1]b = [2; 16; 8; 0; 0; 0] A small sample:Let t_(l),t_(o),t_(m),t_(n),t_(t),t_(s) be starttimes of the associated tasks.Now use the graph to write thedependency constraints:Tasks o,m, and n can't start until task I is finished, and task Itakes 3 days to finish. So the constraints are:t_(l)+3<=t_(o),t_(l)+3<=t_(m),t_(l)+3<=t_(n)Task t can't start until tasks m and n are finished. Therefore:t_(m)+1<=t_(t),t_(n)+2<=t_(t),Task s can't start until tasks o and t are finished. Therefore:t_(o)+3<=t_(s),t_(t)+3<=t_(s)Type in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:$$\begin{aligned}[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 +…Suppose you have been given a map of 6 cities connected with each other via different paths. Your job is to visit every city just once covering the minimum distance possible. Solve this problem using Genetic Algorithm. You can start at any point and end at any point. Just make sure that all the cities have been covered. 1. Encode the problem and create an initial population of 3 different chromosomes 2. Choose any one parent from your above solution and identify the following:i. Gene ii. Chromosome 3. Think of an appropriate fitness function to this problem and give proper justification. 4. Use the fitness function to calculate the fitness level of all the chromosomes in your population. Select the fittest 2 chromosomes based on the fitness function. 5. Perform crossover that you have been taught in the class on the selected parents. Now based on the offspring, for this problem do you think that is the best way to perform crossover? If not, explain why. 6. Perform mutation that you…