Math, Midterm 1

February 22, 2020 MATH 262 Midterm 1 Review Questions 1. Which of the following sets (with natural addition and multiplication by a scalar) are vector spaces? Justify your answers. (a) The set of all continuous functions on the interval [0, 1]; (b) The set of all non-negative functions on the interval [0, 1]; (c) The set of all polynomials of degree exactly n; (d) The set of all polynomials of degree exactly n and the zero polynomial; (e) The set of all polynomials of degree at most n; (f) The set of all symmetric n × n matrices, i. e. the set of matrices A = (aij ), i, j = 1 . . . n such that aij = aji . 2. True or false: (a) Every vector space contains a zero vector; (b) A vector space can have more than one zero vector; (c) An m × n matrix has m rows and n columns; (d) If f and g are polynomials of degree n, then f + g is also a polynomial of degree n; (e) If f and g are polynomials of degree at most n, then f + g is also a polynomial of degree at most n. 3. Suppose v and w are elements of a vector space V . Prove using axioms and showing all steps: (a) (v + w) + w = v + 2w. (b) ( 21 v + w) + 21 v = v + w. (c) −v is unique: if v + w = v + x = 0 then w = x. (d) −v = (−1)v. (e) w − v = −(v − w). 4. Consider the set R2 with the operations (v1 , v2 ) + (w1 , w2 ) = (v1 + w1 − 3, v2 + w2 + 5) r(v2 , v2 ) = (rv1 − 3r + 3, rv2 + 5r − 5). (a) Determine the additive identity (“the zero vector”) and verify that the definition of the additive identity indeed holds for this vector. (b) Given a vector v = (v1 , v2 ) determine its additive inverse −v. (b) Verify the distributive law: r(v + w) = rv + rw. 5. Consider the functions f (x) = cos2 x, g(x) = sin2 x, h(x) = cos(2x), i(x) = 1. (a) Express f (x) as a linear combination of g(x), h(x) and i(x). (b) Express g(x) as a linear combination of h(x) and i(x). (c) Show that there are infinitely many possible linear combinations that could be used in part (a). 6. Determine whether the following subsets of R3 are subspaces: (a) S = {(x, y, z) ∈ R3 : |x| = |y| = |z|}. (b) S = {(x, y, z) ∈ R3 : x ≤ y ≤ z}. (c) S = {(x, y, z) ∈ R3 : x2 + y 2 + z 2 + 2xy − 2xz − 2yz = 0}. 7. (a) Show that the sets E = {f ∈ F (R) : f (−x) = f (x) for all x ∈ R} and O = {f ∈ F (R) : f (−x) = −f (x) for all x ∈ R} (the sets of even and odd functions on R) are subspaces of F (R). (b) Show that any function in F (R) can be represented as a sum of a function in O and a function in E. 8. What is the next step of applying the Gauss-Jordan algorithm to the matrices?   1 2 0   (a) 0 1 −1 0 2 0   1 2 0 5   (b) 0 0 1 −1 0 2 0 2 2   1 2 0 1    (c)   0 0  0 10   1 2 3 4 5   (d) 0 1 0 0 0 . 0 0 1 −1 −2 9. Consider the system of linear equations:    x1 + x2 + 2x3 − x4 = 0 x1 + x3 + 2x4 = 0   −x1 + 2x2 + x3 − 8x4 = 0 (a) Write the augmented matrix for this linear system. (b) Using Gauss-Jordan algorithm, find RREF. (c) Using RREF find the general solution of the system of linear equations. Express the solution as         x1 a v1 w1 x   b  v  w   2    2  2   =   + s  + t . x3   c  v3  w3  x4 d v4 w4 10. If possible represent the polynomials p(x) given below as linear combinations of the following ones: p1 (x) = x + x2 , p2 (x) = 2 + x + x3 , If such representation is impossible give reasons why. (a) p(x) = 1 + x + x2 + x3 (b) p(x) = x3 − x2 + x + 3 (c) p(x) = x3 . 3 p3 (x) = x2 + x3 . Answers: 1. (a) yes; (b) no; (c) no; (d) no; (e) yes; (f) yes. 2. (a) true; (b) false; (c) true; (d) false; (e) true. 3. (c) If v + w = v + x then (−v) + (v + w) = (−v) + (v + x) (axioms (A1) and (A6)), then ((−v) + v) + w = ((−v) + v) + x) (Axiom (A4)), then (v + (−v)) + w = (v + (−v)) + x) (Axiom (A3)), then 0 + w = 0 + x (Axiom (A6)), then w + 0 = x + 0 (Axiom (A3)), then w = x (Axiom (A5)). 4. (a) ~0 = (3, 5); (b) −(v1 , v2 ) = (6 − v1 , 10 − v2 ). 5. (a) f (x) = (−1)g(x) + 0h(x) + 1i(x) is one possibility; (b) g(x) = − 12 )h(x) + 21 )i(x); (c) Since f = i − g and 2g + h − i = 0, then f = i − g + C(2g + h − i), so f (x) = (2C − 1)g(x) + Ch(x) + (1 − 2C)i(x), where C is arbitrary. 6. (a) no; (b) no; (c) yes; hint: x2 + y 2 + z 2 + 2xy − 2xz − 2yz = (x + y − z)2 7. (b) f (x) = 21 (f (x) − f (−x)) +    1 1 2 0    (b) 0 8. (a) 0 1 −1 0 0 0 2 1 (f (x) 2 + f (−x)).   1 2 0 5   2 0 2 0 (c)  0 0 1 −1 0  2 1   0   1 2 0 7 11   (d) 0 1 0 0 0 . 0 0 1 −1 −2 0             −2 0 0 x1 1 0 1 2 0 1 1 2 −1 0 3 −1 x  0      2       9. (a)  1 0 1 2 0; (b) 0 1 1 −3 0; (c)   =   + s   + t   . 0 1 x3  0 0 0 0 0 0 −1 2 1 −8 0 1 0 0 x4 10. p(x) = 21 p1 (x) + 12 p2 (x) + 12 p3 (x); (b) p(x) = − 21 p1 (x) + 32 p2 (x) − 12 p3 (x); (c) impossible: the system of linear equations with the augmented matrix  0 1   1 0 2 1 0 1 0 0 1 1  1 1   1  1 0   0 0 0 1 0 0 0 0 1 0  0 0  . 0 1 1 has no solution since its RREF is 4