Zero Vector - Linear Algebra - Quiz Solution, Exercises of Linear Algebra

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Math 205A Quiz 03, page 1 September 28, 2007 NAME ul 1. Let S be the set of vectors {c, c2,...,¢,}, where each ¢; is a column vector in R™. Some of the following statements are true (ie, they are theorems) and others are not. Circle each letter for which the statement is true, and mark an “X” through the letter if the statement is false. (a) If m < & then 9 is linearly dependent. BAlf & If at least one vector in S is not a linear combination of other vectors in S$, then S is linearly independent. The set S is linearly dependent <=> at least one vector in S can be expressed as a linear combination of the others. If the only solution to Ax = 0 is the trivial solution, where the members of S form the columns of A, then S is linearly independent. 2. By inspection, classify each of the following sets as linearly independent (LI) or linearly dependent (LD). Justify your answer by writing the letter corresponding to the theorem from problem 1 next to your answer of “LI” or “LD”. For example, if you think the set in 2Q is linearly independent and statement j in problem 1 is your reason, then you'd write “LI-j” as your answer. ab 2: 3 4 | | | A | ol jal isl la This matrix & tn ,|-7], [O]}, |o 2B. ol: lol: lsl- fa echelon Bee aud BL of} fo} fof fal) evs evry ceabime si €=-3) & pivot celoma. Np free EE) fol fie vale => Al = 6 hae {| tr [°77]} ap. ¢ |3} J} {°) | 7? Only the tavink solodom 7 cosm/4]? [e°—1 “Viale| sy yap ps7 thus tte Cole, are Le 4 4 9 3 * Berge véclyr e@D* “th chear Haat Greg? Sy ro 23 od 10 i 3A: Express c; as a linear combination ¢ of other column vectors or explain why you can’t. THK Clear Hot. 6, +& seu y $0 % = Fy-S, 3B: Express cz as a ines combination of other column vectors or explain why. you can’t. e Hh my Any Le Xo, 4 Kos + Xq cy ben te tem TG o] but 22. i werd sok amt a Le ga %2 Ba. 3C: This example should explain why one of the statements in problem 1 above is false. Which ~ (a) since. A consis a) K=4 column veetors ean f which sa R™ adh m= 3, ve hae mek So by '@, Phine columns at LP TF A) baie tove, every Lalymn woul bea LE o te others, get 3B shou, Hui not: fo be the care, The rahe abo daus @ é take 3 either (i) on @) On, fa Phe, Gnsuee to fi porvlom ,