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Linear Algebra 2270 Midterm 1 group B Axioms of real numbers P: For any x, y, z E IR: RA1) x+y=y+x RA2) (x+y)+z=y+(x÷z) RA3) there is a natural element of addition denoted by 0, such that for any x E l: x+0=x RA4) there is an additive inverse —x such that x + (—x) = 0 RA5) xy=y•x RA6) (xy).z=y.(x.z) RA7) there is a natural element of multiplication denoted by 1, such that for any x € : x1 =x RA8) If x * 0, there is a multiplicative inverse RA9) x(y+z)=xy+x.z such that x () = 1 Solve four out of six problems below. If you attempt to solve more than four problems, mark the ones that you want to be graded. 1. Intersections in England are olten constructed as one way “roundabouts,” such as the one shown in the figure. Assume that traffic must travel in the directions shown. Find the general solution of the network flow. Find the smallest possible value for x . 5 B I O() 2. Check if the following vectors are linearly independent or not. If they are dependent, find a linear combination that is equal to ©, but has not all coefficients equal to 0. 0 = 2 1 v 1 2 V 1 —1 2 , v = 2 3 1 3 3. Row reduce the following augmented matrices and write the solution set in the parametric form, i.e. find vectors q, qi,. q € RT such that the solution set is . . {zi=qo+aq+...+akqk: 0 1 1 —1 1 2 0 —3 1 1 2 0 —1 6 —1 8 a1,...,ak€l} I I —3 1 5 (1) 4. What is the definition of a vector space? 5. Let V be a vector space and fV its subspace. Is the set: A = {5w : w E W} a subspace? If you think it is, prove it. If not, give an example of U, W for which A is not a subspace. 6. Consider a space of functions UR. Using the axioms of real numbers and the definition of vector addition and scalar-vector multiplication in , prove that for any f, g, h E fiR, (f+g)+hrf+(g+h) Do one step at a time and state clearly what axiom or definition you use at each step. 2 LL 1-1 C 0 0 N -i-- 0 -4- 0 - 0 cr— c L- 1- 0 •c 1 >L 0 0 0 C) 00 C 0 c (N 0 000 0 oo 0 ‘I C — c— —0000 %,Q —fl , I C) J c, x ‘ L - < ‘) ‘I < )< < 1 c-) -4 4-, - C) ‘4- - X 2 - + N ‘f 7 -I- 0 r4 1’ I’ H “1 -f 0 c, , - — r’j p - ( I f&?[L 0 z 0 C 0 - I - -‘ 3 I il —1 0 ° HJ -I 0 -3 25 o —1 t2 2 o o r10 C’ 0 j t 0 L° o J x -G - - cCe 5 x — 4Q = - j 3 iiviio’t(cs. --13 2 SCLA” e-- 10 tc 1’ U xI Li 1—i;) N r I---! 4’ I I L; ( HI J 0 [ I i I j +: 4 I ‘5 o —3 2 1 0 -2-a 00/ 3 x C€L 0 / 2 (3: 4, 1 0 -I o 3I ‘1 - I o Is_fl ‘ o I o 2 0 L _31 -I 2 I £ f(H)r 3 1 -;: 1] £ r B.4 A vector space V over the scalar field is a set with two operations “+“ and Operation “+“, called vector addition, takes two elements of V and returns one element of V. Operation called scalar-vector multiplication takes one real number and one element of V and returns one element of V. The operations satisfy the following conditions: “•“. “•“, Axioms of a vector space V: arbitrary u, v, w c V and for arbitrary c, d E O we have u+v=v+u (u+v)+w=u+(v+w) there exists a natural element of addition 0 c IR, which satisfies u there exists an additive inverse (—u), such that For Vi) V2) V3) V4) V5) V6) V7) V8) c.(u+v)=c.u+c.v (c + d) = c u + d• u c.(d.u)=(c.d).u iu=u . Elements of V are called vectors and real numbers are called scalars. i + U = u L ) ‘IJ ct C’ > C’ :3 = - . ‘ . ,, 9 . . 3 3 .4’ Ii 3 --.. 3’ i L) 2‘I 5 cL sj .1) \1’ -3 3 -N -? L cj cj II 3 a 2 -, 2 Li” •JI c) 0 ‘-1 - § -4- 4 - — - c) - —.-- 5, S 0-, -4- 3, 0’ 3, L o o - -. & — - 1- p - - —‘ o- *;f ‘;.J -S I -5- —‘ -s ‘—- -4 ‘I -- 3, ci L - 1- —S O) flj O) -- - -4- vTh - —.‘ ‘—_d. —.‘ -r - +