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Linear Algebra 2270 Midterm 1 group A Axioms of real numbers [R: For any x, y, z E tR: 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 x+0=x RA4) RA5) RA6) RA7) there is an additive inverse —x such that x + (—x) = 0 x•y=y•x (x.y).z=y.(x.z) there is a natural element of multiplication denoted by 1, such that for any x E fi: x1=x RA8) If x * 0, there is a multiplicative inverse RA9) x.(y+z) =xy+xz 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. Balance the following chemical reaction (i.e. find numbers to be put in the place of the number of Al, 0 and C atoms are the same on both sides of the reaction): -- 3 0 2 A1 + -- C -* -- Al + -- ““ such that 2 CO Hint: For each compound construct a vector that lists the number of atoms of Al, 0, C. And then write a linear system with one equation for atom. 2. Check if the following vectors are linearly independent or not. If they are dependent, find a linear combination that is equal to U, but has not all coefficients equal to 0. 0 = 2 1 v 1 , 1 2 1 V2 , V3= —1 2 5 3. Row reduce the following augmented matrix and write the solution set in the parametric form, i.e. find vectors qo, q,. . , q c fifl such that the solution set is . {u=qo +aiqi 1 0 2 —2 —2 —2 +... 1 1 0 +aq: 1 —1 1 (1) al,...,ak c —1 4 5 I I 2 1 —1 4. Write definitions for the following: (a) Let V be a vector space. Let v ,.. 1 dent? , v e V. What does it mean for vl,. .. , v to be indepen (b) How is the addition and scalar vector multiplication defined in the space of functions = 1 {f : -* 5. For sets A, B, the sum of those two sets A u a’ c Au B B is defined as — a’ e A or a’ e B Let V be a vector space and TV . TV 1 2 its two subspaces. Is it true in general that TV 1 u TV 2 is a subspace. If you think it is true, prove it. If you think it is not true, give a example of V 1 TV TV , 2 for which it is not true. that in the original version of the problem there was an error, which has been spotted during the exam. The error was that ‘or” was replaced by “and”) (J\Totice 6. Consider a space of functions l. Using the axioms of real numbers and the definition of vector addition and scalar-vector multiplication in , prove that for any f e fi, c, d E c.(d’f) = (c.d)’f Do one step at a time and state clearly what axiom or definition you use at each step. 2 C - - ‘?c> t fj -, >c S c c3- C - ¶ — :: Ii Lo c 0 cjJI— V I-i cS. If + ccs >c I, I) — r) ‘I ‘Jj ‘I - Th F’.-) ç C r.- 5) P + cç cjJ 10 17 3) - 23 ) 9ç L 1iI 7ya?tj, r - -o c )—) 1’17 . 0 0 7 Q 1 1 ‘ 1 J oJ J(>’J {I - / (* ii 11) cr1 ztl III ‘I 0 •1 I, -.7 — — ci 0 — -f I 0 ‘— N N 0 c ‘7 c) c) -. 0- jTh 1 p—) -- 0 0 r”J 1 N - I N 0 r\J I) + — f-J-t Li I C ‘p V 2 Vi i’ a Ls- f4e LJ, itd 2 t? cct9ace V Wa I LJ: 6l wV 44 V Lc I I LJ( So LiQ l,’Jz (A6 ) t)i i’-’’1 )’kiz 1 V6l) t1o1 (A4V £- I 4 a C( i 1Qi ‘ i’- i 17oF € ‘spoce /i?Y. 0 -ç 4 cm p (1 > - 0 4N1 Lc (N - I,’