Q4 (4 marks) Hand in c), d) only. Which of the following are vector subspaces of...
Question 1: Vector Spaces and Subspaces (a) Show that (C(0, 1]), R, +,), the set of continuous functions from [0, 1 to R equipped with the usual function addition and scalar multiplication, is a vector space. (b) Let (V, K, +,-) be a vector space. Show that a non-empty subset W C V which is closed under and - necessarily contains the zero vector. (c) Is the set {(x,y)T: z,y E R, y a subspace of R2? Justify.
10. Determine the values of k for which the system of linear equations has (i) no solution vector, (ii) a unique solution vector, (iii) more than one solution vector (x, y, z): (a) kx+ y+ z= (b) 2x + (k-1)y + (3-k)2-1 2y + (k-3): = 2 x+ky + z = 1 -2y+ x 2x + ky- z =-2 (c) x + 2y + k= 1 (d) -3z =-3 10. Determine the values of k for which the system of...
EXERCISE 2 [2.5/10] Given the following vector subspaces: W, Ξ {(x, y, z) E R3 / 0) x + y a) [1.0/10] Calculate bases of Wi and W2. b) [1.0/10] Calculate a basis of W1 n W2 c) [0.5/10] Calculate a basis of W1 + W2 EXERCISE 2 [2.5/10] Given the following vector subspaces: W, Ξ {(x, y, z) E R3 / 0) x + y a) [1.0/10] Calculate bases of Wi and W2. b) [1.0/10] Calculate a basis of...
linear algebra 2. Which of the following subsets of Rare actually subspaces? Justify your answer in terms of the definition and properties of subspaces. (a) The vectors [x y z]" with x + 2y -z = 0. (b) The vectors [a b c]" with a + b + c = 3. (c) The vectors [a+2bb-3b]' where a, b are any real numbers, (d) The vectors [pr] where q.r are any real numbers and p20.
TOTAL MARKS: 25 QUESTION 4 (a) Find a normal vector and an equation for the tangent plane to the surface at the point P: (-2,1,3). Determine the equation of the line formed by the intersection of this plane with the plane z = 0. 10 marks (b) Find the directional derivative of the function F(r, y, z)at the point P: (1,-1,-2) in the direction of the vector Give a brief interpretation of what your result means. 2y -3 [9 marks]...
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real valued functions on on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W-f eV: f(0)-0is a subspace of V or not? (b) Verify, if the set W-Uev f(0) 1 is a subspace of V or not? (c) Verify, if the set W-İfEV:f(x)-0V-2-z is a subspace of V or not? 1b) PrtScn Home FS F6 F7 F8 5
part a and b PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following sets W are subspaces of the given vector spaces: (A) The set W to be of all triples of real numbers (x, y, z) satisfying that 2x - 3y + 5z = 0 with the standard operations on Ris a subspace of R3. (B) The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication.
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the point (A) v3 (E) 2v3 (B) 1+2V2 (C) 2 v3 (G) 3/2 (D) V2 6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the...
Can u please answer the question (G) 1. (15 marks total) Consider the real vector space (IR3, +,-) and let W be the subset of R3 consisting of all elements (z, y, z) of R3 for which z t y-z = 0. (Although you do not need to show this, W is a vector subspace of R3, and therefore is itsclf a rcal vector space.) Consider the following vectors in W V2 (0,2,2) V (0,0,0) (a) (2 marks) Determine whether...
MATLAB QUESTION!!! 9) The following system of equations is used in Navier Stokes theorem to determine force fluctuations on a hypersonic jet. Solve the following system of equations using inverse matrix method and left division. Show how long this process takes to compute using Matlab. 3x+y+z+w+3v=27 5v-3y+x+7z+w=17 2x+2y-3z+4w-v=16 x+y+z+w+y=1 y-z+x-w-7v=-7