Question

Lalala Caca 4 photos Problem 1. (26 pts.) SHORT ANSWERS (a) Circle one: If Rn contains a linearly independent set with exactly 15 vectors that is NOT a basis, then n is AT LEAST 15 / AT LEAST 16 AT MOST 14 / AT MOST 15 (b) Circle one: If R" contains a spanning set with eractly 15 vectors that is NOT a basis, then n is AT LEAST 15 / AT LEAST 16 AT MOST 14 / AT MOST 15 (c) Circle one: If the columns of Anxs are linearly independent, then Ar 0 1 / x-many solutions 0 has (d) Circle one: If the rows of Assare linearly independent, the dimension of the null space of Ais 1/2 3 //5 (e) Circle one: If the column space of Axis cractly a two-dimensiunal plane then the null space of As exactly ONE POINT ONE LINE ONE PLANE/3-SPACE/X-SPACE 12:25 PM () Circle one: If the null space of Aexs is exactly a plane, then the column space of A is exactly ONE POINT / ONE LINE / ONE PLANE / 3-SPACE / 4-SPACE

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Answer 1

I Problem 1 (a) non contains al. To set witry exactly 15 vectors that is not a basis we know every l. I. peto com be extended to form cu baris so n 2 15 Answen! TAT LEAST 15 (b) IR Contains a spanning bet with exactly 15 rectors that in not a basis know that spanning set can be reduced to form a basis so n = 15 Answer: At Most 15 (c) Columns of Ayxa ane l. I. no Bra) = 4 Ax=0 where a =(2, 22, 23, 24 .I unknowns=4 so f(A)= no. of unknowns = 4 so unique solution, > Tank Answer. O (d) A3x5 rown y A are l.IS(A) = 3 sve know' by rank- nullity theorem , S(A)+n/A) = 5 = n(A)= 5-2=2= dimn of null spacey A Answer : 12 (e) Aux 3, di columns bace) = 2 = dim ( vorge space) so 81A)= dim ( range space) =2 . so rank- hullity theorem Platn(A)= 3 S => Rial- 3 - 2 - 1 Answer: since Nail space of AS, R3&n141=1 -re. NIA ? (n. 12, 23): 2, 22, M3 ER} is generated by single point. so it in a live só Toner het l o blame 16) Auxg: R3 > JR4, N1A)= { ax: +b2 + c ay teel0, a, b, c, d ER } .. x.x2 are chosen independently . Az in dependent I. so n (A)=2, SCA1= 3-nial= 1 & dim" (Column & pace) Answen! One fine] (i Column Apace (A) {K (2, 3, 4) KER? Escanned with Camscattirer {K, B, 413 in basin.