Extra Practice Solutions1. Describe geometrically the following sets and find a simplified vector equation for each.(a)S1= Span⇢12(b)S2= Span8<:2412335,2423435,24111359=;(c)S3= Span8>><>>:266411003775,2664001137759>>=>>;Solution.(a) A vector equation forS1is~x=c112, c12Rwhich represents a line through the origin inR2.(b) Since2411135=2423435-2412335we have thatS1= Span8<:2412335,2423435,24111359=;= Span8<:2412335,24234359=;Since2412335and2423435are not scalar multiples of one another, a simplified vector equation forS2is~x=c12412335+c22423435, c1, c22Rwhich represents a plane through the origin inR3.(c) Since the set8>><>>:266411003775,2664001137759>>=>>;contains two vectors that are not scalar multiples of one another, a simplified vector equationforS3is~x=c1266411003775+c2266400113775, c1, c22Rwhich represents a plane through the origin inR4.Page 1
2. Determine which of the following sets is linearly independent. If it is linearly dependent, write oneof the vectors as a linear combination of the others.(a)⇢12,13,14(b)8<:2412-335,2461235,24000359=;(c)8<:2411035,2410135,24011359=;Solution.(a) Forc1, c2, c32R, considerc112+c213+c314=00We obtain the system of equationsc1+c2+c3=02c1+3c2+4c3=0We see from the first equation thatc3=-c1-c2and thus the second equation becomes2c1+ 3c2+ 4(-c1-c2) = 0, that is,-2c1-c2= 0. There are infinitely many solutions to thisequation. In particular, if we letc2= 2, thenc1=-1 and it follows thatc3=-(-1)-2 =-1.