Absolute Mad Lads - Raspux 00 PROBLEM SET 02: Vector Algeb x

(1 point) All vectors and subspaces are in R^n Check the true statements below: \text { A. In the Orthogonal Decomposnion Theorem, each lerm } \hat{y}=\frac{y \cdot u_{1}}{u_{1} \cdot u_{1}} u_{1}+\ldots+\frac{y \cdot u_{p}}{u_{p} \cdot u_{p}} u_{p} \text { is itselt an orthogonal projection of } y \text { onto a subspace of } W \text { B. If an } n \times \text { p matrix } U \text { has orthonormal columns. then } U U^{T} x=x \text { for all } x \text { in } R^{n} C. The best approximation to y by elements of a subspace W is given by the vector y- projw (y). D. If y = 21 + z, where z, is in a subspace W and z2 is in W^1, then 2, must be the orthogonal projection of y onto W. E. Ir W is a subspace of R^n and if v is in both W and W, then v must be the zero vector.

Suppose that the trace of a 2 x 2 matrix A is tr(A)=-2 and the determinant is det(A) = -63 Find the eigenvalues of A. The eigenvalues of A are __________ (Enter your answers as a comma separated list.)

Consider the following for the parametric equations below. a. Eliminate the parameter to find a Cartesian equation of the curve. b. Sketch the curve using a table, and indicate with an arrow the direction in which thecurve is traced as the parameter increases. \text { c. Find } \frac{d y}{d x} \text { and } \frac{d^{2} y}{d x^{2}} \text {. } x=1+e^{2 t}, y=e^{t}

2.(a) (i) Use row operations to find the inverse of (ii) Using the inverse matrix found above, solve the equation AX = B where (b) Use properties of determinants to (i) Factorise A=\left[\begin{array}{ccc} 1 & -1 & 2 \\ 3 & 2 & 4 \\ 0 & 1 & -2 \end{array}\right] B=\left[\begin{array}{l} 3 \\ 6 \\ 1 \end{array}\right] \left|\begin{array}{ccc} z^{2} & y^{2} & x^{2} \\ z & y & x \\ 3 & 3 & 3 \end{array}\right| \left|\begin{array}{ccc} 1 & b c & b c(b+c) \\ 1 & c a & c a(c+a) \\ 1 & a b & a b(a+b) \end{array}\right|=a b c\left|\begin{array}{lcc} a & 1 & b+c \\ b & 1 & c+a \\ c & 1 & a+b \end{array}\right| \begin{aligned} x+y-3 z+w &=& 2 \\ 2 x-y-3 z-2 w &=&-5 \\ x-3 y+z+4 w &=&-10 \\ 3 x+4 y-10 z-7 w &=& 9 \end{aligned} (ii) Prove, without expanding that (c) (i) Represent the following system of equations as an augmented matrix. (ii) Reduce the matrix in (c)(i) to row echelon form, find the rank and hence show that the system of equations admit an infinite number of solutions.Find a general solution for this system.

\text { a) The matrix } Y=\left[\begin{array}{lll} 0 & 1 & 2 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{array}\right] \text { has } 0 \text { as its only eigenvalue. } Find the general solution to the problem of finding the corresponding eigenvectors. \text { The matrix } Z=\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right] \text { has } 1 \text { as its only eigenvalue } Find the general solution to the problem of finding the corresponding eigenvectors.

2. Three-dimensional rotation of vectors can be descried by 3 × 3 matrices. A rotationaround the x-axis by angle 0 is given by the matrix O_{x}(\theta)=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \end{array}\right) A rotation around the y-axis by angle ø is given by the matrix O_{y}(\phi)=\left(\begin{array}{ccc} \cos \phi & 0 & \sin \phi \\ 0 & 1 & 0 \\ -\sin \phi & 0 & \cos \phi \end{array}\right) and a rotation around the z-axis by angle x is given by the matrix O_{z}(\chi)=\left(\begin{array}{ccc} \cos \chi & \sin \chi & 0 \\ -\sin \chi & \cos \chi & 0 \\ 0 & 0 & 1 \end{array}\right) a) The vector V is given by \mathbf{V}=\left(\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right) Show by explicit calculations that O_{z}(\chi=\pi) O_{y}(\phi=\pi / 2) \mathbf{V}=O_{y}(\phi=-\pi / 2) \mathbf{V} Explain with words the geometric interpretation of this equation. Evaluate the composite rotation O2(x1)Oz(x2) and express it as a rotation around the z axis with respect to a single angle. By using the iteration procedure, determine0:(x), where n is a positive integer, in terms of a rotation with respect to a single angle.[6] Find the general angles x and phi of two successive rotations of the form O2(x)O,(phi)acting on vector V such that the resulting vector corresponds to a rotation Ox(0)of V around the x-axis for a given angle phi. Draw the corresponding diagram that depicts the rotations of V with respect to the angles theta, phi and x.[8]

A two-dimensional, Incompressible flow is given by u = - y and v = x. Show that the streamline passing through the point x = 10 and y = 0 is a circle centered at the origin. x^2 + y^2 =_________. exact number, no tolerance

3. (a) Sketch and fully describe the surface Sph given, in terms of Cartesian coordinates (x, y, z), by Sph = {(x, y, z) : (x − xo)² + (y − yo)² + (z − zo)² = a²}, where a, xo, yo and zo are constants, and show that it can be parameterized by r(0,0) = (xo + a sin cos , yo + a sin sind, zo + a cos 0), giving the ranges of the parameters and p. (b) For the surface Sph, defined in part (a), show that a vector surface element is given by ds = a sin 0 [r - (To, yo, 2o)] dedo, and justify the sign convention associated with this normal vector. Hence evaluate the flux ƒ= $₁₂ Sph G.dS of the vector field G = (1,0, z²) out of the surface. (c) Using your answer to part (b), evaluate L = lima-03f/(4πa³). (d) State the (coordinate-invariant) integral form of the definition of divergence. By referring to your answer to part (c), demonstrate that it is consistent with the Cartesian definition of divergence in this case. (e) By applying the Divergence Theorem to the vector field u = ow, where is an arbitrary differentiable scalar field and w is a constant vector, derive the identity JI føds, where the closed surface S is the boundary of the volume V. VodV=

For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given by the equation v=9 x y-x^{2} y Determine the velocity component in the x direction so that the volumetric dilatation rate is zero. Denote the x-independent constant as f(y).

4. The triple scalar product is defined by It is known that if the triple scalar product of 3 vectors is equal to zero then the3 vectors are coplanar (i.e. all 3 vectors are on the same plane). \text { a) Show that the } 3 \text { vectors } \vec{u}=[1,2,-3], \vec{v}=[0,1,1] \text { and } \vec{w}=[2,1,-9] \text { are coplanar. } b) Using geometrical reasoning, explain why if ū, v, and w are coplanar, then ū × v · w = 0.