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MACS-332A - Linear Algebra Homework #1: 1.1 - 1.3 True or False. You must justify your answer. 1. Elementary row operations on an augmented matrix never change the solution set of the associated linear system. True - Information preceding the subsection, Existence and Uniqueness Questions, pg 8. 2. Two linear systems are equivalent if they have the same solution set. True - This definition is in the second paragraph after equation (2), pg 3. 3. The echelon form of a matrix is unique. False - Only the reduced echelon form is unique, pg 15. 4. Whenever a system has free variables, the solution set contains many solutions. False - The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty, pg 28. 5. When u and v are nonzero vectors, Span{ u, v } contains the line through u and the origin. True - See Fig. 11, pg 35. 6. Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span{a1 , a2 , a3 }. True - See the paragraph following the definition of Span, pg 35. 1. Determine the quadractic polynomial p(t) = a0 +a1 t+a2 t2 , which interpolates the data (1,12), (2,15), (3,16). That is, determine a0 , a1 and a2 such that the following equations: a0 + a1 (1) + a2 (1)2 a0 + a1 (2) + a2 (2)2 a0 + a1 (3) + a2 (3)2 are satisfied. = = = 12 15 16 1 1 1 12 The corresponding augmented matrix is 1 2 4 15 1 3 9 16 1 1 1 12 3 Solving this system using row reduction yields the echelon form: 0 1 3 0 0 1 −1 We can see that this system is consistent. Rowreducing this to a reduced echelon form yields (with 1 0 0 7 1 0 6 . Thus the polynomial is p(t) = 7 + 6t − t2 . the pivot positions circled): 0 0 0 1 −1 1 2. Given the system x1 + 3x2 3x1 + hx2 = = 2 , choose h and k such that the system has k (a) Is consistent with a unique solution. (b) Is consistent with infinitely many solutions. (c) Is inconsistent. Augmented Matrix: 1 3 2 3 h k ⇒ Echelon form: 1 3 2 0 h−9 k−6 (a) For h 6= 9, the system is consistent with a unique solution. (b) For h = 9 and k = 6, the system is consistent with infinitely many solutions. (c) For h = 9 and k 6= 6, the system is inconsistent. 3. Determine if b is a linear combination of the vectors formed from the columns of the matrix A. 5 3 22 7 b = 20 A = −4 9 −2 15 This is equivalent whether thelinear to asking system withaugment matrix [a1 a2 b] has a solution. 5 3 22 5 3 22 5 3 22 188 188 47 −4 7 20 → 0 → 0 47 5 5 5 5 37 123 0 −5 − 5 0 0 5 9 −2 15 This system is inconsistent. Therefore b cannot be expressed as a linear combination of the columns of A. 4. Suppose a, b, c and d are constants such that the system ax1 + bx2 cx1 + dx2 = = f , g with a 6= 0 is consistent for all possible values of f and g. Determine a single equation which restricts the numbers a, b, c and d. a b f a b f Augmented Matrix: ⇒ Echelon form: c d g 0 − ac b + d − ac f + g For this matrix to be consistent for all possible f and g, it must be the case that c − b + d 6= 0 ⇒ ad − bc 6= 0. a 2