Linear Algebra

Section I. Solving Linear Systems 9(a) 2x + 2y = 5x − 4y = 0(b) −x + y = 1x + y = 2(c) x − 3y + z = 1x + y + 2z = 14(d) −x − y = 1 (e) 4y + z = 20 (f) 2x + z + w = 5−3x − 3y = 2 2x − 2y + z = 0y − w = −1x + z = 5 3x − z − w = 0x + y − z = 10 4x + y + 2z + w = 9̌ 1.18 There are methods for solving linear systems other than Gauss’ method. Oneoften taught in high school is to solve one of the equations for a variable, thensubstitute the resulting expression into other equations. That step is repeateduntil there is an equation with only one variable. From that, the first numberin the solution is derived, and then back-substitution can be done. This methodtakes longer than Gauss’ method, since it involves more arithmetic operations,and is also more likely to lead to errors. To illustrate how it can lead to wrongconclusions, we will use the systemx + 3y = 12x + y = −32x + 2y = 0from Example 1.12.(a) Solve the first equation for x and substitute that expression into the secondequation. Find the resulting y.(b) Again solve the first equation for x, but this time substitute that expressioninto the third equation. Find this y.What extra step must a user of this method take to avoid erroneously concludinga system has a solution?̌ 1.19 For which values of k are there no solutions, many solutions, or a uniquesolution to this system?x − y = 13x − 3y = ǩ 1.20 This system is not linear, in some sense,2 sin α − cos β + 3 tan γ = 34 sin α + 2 cos β − 2 tan γ = 106 sin α − 3 cos β + tan γ = 9and yet we can nonetheless apply Gauss’ method. Do so. Does the system have asolution?̌ 1.21 What conditions must the constants, the b’s, satisfy so that each of thesesystems has a solution? Hint. Apply Gauss’ method and see what happens to theright side. [Anton](a) x − 3y = b 1 (b) x 1 + 2x 2 + 3x 3 = b 13x + y = b 2 2x 1 + 5x 2 + 3x 3 = b 2x + 7y = b 3 x 1 + 8x 3 = b 32x + 4y = b 41.22 True or false: a system with more unknowns than equations has at least onesolution. (As always, to say ‘true’ you must prove it, while to say ‘false’ you mustproduce a counterexample.)1.23 Must any Chemistry problem like the one that starts this subsection — a balancethe reaction problem — have infinitely many solutions?̌ 1.24 Find the coefficients a, b, and c so that the graph of f(x) = ax 2 +bx+c passesthrough the points (1, 2), (−1, 6), and (2, 3).