APPM 2360 Midterm: appm2360fall2016exam2_sol

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turquoisegnu128

31 Jan 2019

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Problem 1: (24 points) true/false (answer true if it is always true; otherwise answer false) No justi cation is needed. (a) the system of equations (cid:20)a b d e f(cid:21) ~x = (cid:20)1. 5 are vectors, and w = span{~v1, ~v2, ~v3, ~v4, ~v5}, then the dimension of w is always 5. (d) the set of all 4 4 matrices that have 0 entries on the diagonal is a vector space. The dimension of w can be 0,1,2,3,4 and 5. (d) true, it satis es the closure properties under addition and scalar multiplication. x2 x 1 = (cid:0)x2 + 1(cid:1) (x + 2) . (1) Is this set a vector space? (remember to brie y justify your answer). (c) consider the set. For example, y(t) = 0 is not a solution, so it cannot be a vector space. (c) (i) k 6= 0 makes this set of three vectors linearly independent.

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8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

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ceruleangoose957

3. (7 points) Decide whether the following statements are trueor false (give a short justication if you
want full points!):
(a) A homogeneous system of three equations and four variables hasa nontrivial solution.

(b) If u1; u2 are both solutions to the system Ax = b then(u1+u2)/2 is also a solution.

(d) If A;B;C are nxn non-singular matrices, then (ABC)^-1 =C^-1A^-1B^-1

(e) There can be a 4x4 non-singular matrix A such that Ax = 0 has anon-trivial solution (i.e.
x is a solution with some non-zero entries).

(f) Every square matrix with 1's down the diagonal isinvertible

(g) If A is invertible, then A^2 is invertible too.

APPM 2360 Midterm: appm2360fall2016exam2_sol

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