1. Letū,7, andū be arbitrary non-zero vectors in 3-dimensional space. Determine which of the following best...
3.[4p] (a) In the following questions assume that a linear operator acts from a finite- dimensional linear space X to X, and assume that the word "vector means an element of X. Recall that a vector a is a pre-image of a vector y (and y is the image of x) for a linear operator A: X -> X, if Ax-y. How many of the following statements are true? (i) A linear operator maps a basis into a basis. (ii)...
3. [4p (a) In the following questions assume that a linear operator acts from a finite dimensional linear space X to X, and assume that the word "vector" of X. Recall that a vector x is a means an element pre-image of a vector y (and y is the image of x) for linear operator A: X -> X, if Ac y. How many of the following statements are true? a (i) For any linear operator every vector is co-linear...
Exercise Set Chapter 3 Q1) Let u = (2, -2, 3), v = (1, -3, 4), and w=(3,6,-4). a) Evaluate the given expression u + v V - 3u ||u – v| u. V lju – v|w V X W ux (v x W) b) Find the angle 8 between the vector u = (2,-2,3) and v = (1, -3,4). c) Calculate the area of the parallelogram determined by the vector u and v d) Calculate the scalar triple product...
Problem #7: Which of the following statements are always true for vectors in R3? (i) If u (vx w)-4 then w - (vxu)-4 (ii) (5u + v) x (1-40 =-21 (u x v) (ili) If u is orthogonal to v and w then u is also orthogonal to w | V + V W (A)( only (B) (iii) only (C) none of them (D) (i) and (iii) only (E) all of them (F) (i) only (G)i and (ii) only (H)...
2. (-/1 Points] DETAILS POOLELINALG4 6.1.003. MY NOTES Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail to hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.) The set of all vectors [] in R2 with xy > 0 (i.e., the union of the first and third quadrants),...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [ 3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ] (i) Calculate A + B, (ii) Calculate AB (iii) Calculate the inverse of B, (iv) Calculate the determinant of C. (b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2) and (5, −1, 4) respectively. (i) Show that P Q~ =...
You are given that a 4-dimensional pseudo-Riemannian space-time has the interval ds2dudvf (u) dx2 g?(u) dy*, (u, v, x, y) in terms of the coordinates x^ = (i) Use the standard variational principle 2 ds dt = 0 dt ti to find the r-equation governing the geodesic, with parameter t, between given points t and t2 (ii) Deduce from the x-geodesic equation obtained in (i) that f' T.. T. =. ur f where a prime denotes differentiation with respect to...
- In each part, determine whether the pairing (, ) determines an inner product on the vector space V. Justify your answer. (1) V=R>, <[ 3 ][ ==x". (ii) V = R", (7,w) = tr((AU) Aw) where t denotes transpose and A is an invertible matrix. (iii) V =R”, (7,w) = tr((AT)* Aw) where t denotes transpose and A is a non-invertible matrix. (iv) V = C([0,1],R), (f(x),g(x)) = 5o+ f(x)g(x)dx.
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...