


(yi,U2,U3) İn R3 define an inner (x1, T2,23) and y 4. Decide which of the suggested operations on x product (a) x, y 22...
170. A Different Looking Inner Product. Verify that the operation (x,y) = x1y1 - 2192 - 1241 +3.242 where x = (x1, x2) and y = (91, y2) is an inner product in R2. 171. General Inner Products. Decide which of the suggested operations on x = (21,12,13) and y = (y1, y2, y3) in R3 define an inner product: (a) (x,y) = 141 + 2x2y2 + x3y3, (b) (x,y) = xiyž + x3y2 + 3y, (c) (x,y) = x1y1...
(Higher-order linear differential equations) (a) Show that yi (x)-z?, уг (z)-r3, and U3(z) = 1/x are linearly independent solutions of 3. хзу",-z?y"-2xy' + 6y-0 on (-oo, 0) and (0, +00). Write down the general solution to (4 (b) Find a fundamental set S of solutions of
(Higher-order linear differential equations) (a) Show that yi (x)-z?, уг (z)-r3, and U3(z) = 1/x are linearly independent solutions of 3. хзу",-z?y"-2xy' + 6y-0 on (-oo, 0) and (0, +00). Write down the general...
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
3. Let R3 be equipped with the inner product (x,y) = Ax. Ay, where A is the matrix shown below: TO A=13 LO -4 2 0 2 1 5) a.) (5 points) Let v = (1,-1,3). Find ||v||. b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer.
3. Let R3 be equipped with the inner product (x, y) = AX Ay, where A is the matrix shown below: TO -4 21 A = 3 2. LO 0 5) a.) (5 points) Let v = (1,-1,3). Find || V ||. UN b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
EXERCISE 57 (2-27). Define g, h: {x E R2· 1} → R3 by g(x, y) = (x, y, V1-x2ー),2) , h(x, y) = (x, y,-V1-x2-y2) Show that the maximum off on {x E R3 : 11x11 = 1} is either the maximum of fo g or the maximum offo h on {x e R2· 1} .
4. (25 points) Which of the following subsets of R3 are subspaces.Explain. a) {(x, y, z) 1 x 0, y 0, z ? c) {(z, y, z) | x2 + y2 + z2-1} d) Is the set H of all matrices of the form |(a,0)T, (b,d)T] a subspace of the space of all 2x2 matrices with the usual matrix addition and scalar multiplica- tion?
In space R^3, we define a scalar product by regulation 〈(x1, y1, z1), (x2, y2, z2)〉 = 2x1x2 + y1y2 + 2z1z2 + x1z2 + x2z1. (a) [10] Calculate the perpendicular projection of the point T (1, 1, 1) on the plane U in R3 with the equation x + 2y + 2z = 0 with respect to the given scalar product. (b) [10] Let φ: R^3 → R be a linear functional with φ (x, y, z) = x...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B.
4. Consider the vector space...