Linear Algebra

Section I. Definition of Vector Space 77sum ( ) ( ) ( )v 1 w 1 v 1 + w 1+ =v 2 w 2 v 2 + w 2is a column array with two real entries, and so is in R 2 . For (2), that additionof vectors commutes, take all entries to be real numbers and compute( ) ( ) ( ) ( ) ( ) ( )v 1 w 1 v 1 + w 1 w 1 + v 1 w 1 v 1+ === +v 2 w 2 v 2 + w 2 w 2 + v 2 w 2 v 2(the second equality follows from the fact that the components of the vectors arereal numbers, and the addition of real numbers is commutative). Condition (3),associativity of vector addition, is similar.(()v 1+v 2( )w 1) +w 2( )u 1=u 2()(v 1 + w 1 ) + u 1(v 2 + w 2 ) + u 2()v 1 + (w 1 + u 1 )=v 2 + (w 2 + u 2 )( ) (v 1= + (v 2)w 1+w 2( )u 1)u 2For the fourth condition we must produce a zero element — the vector of zeroesis it. ( ) ( ) ( )v 1 0 v 1+ =v 2 0 v 2For (5), to produce an additive inverse, note that for any v 1 , v 2 ∈ R we have( ) ( ) ( )−v 1 v 1 0+ =−v 2 v 2 0so the first vector is the desired additive inverse of the second.The checks for the five conditions having to do with scalar multiplication aresimilar. For (6), closure under scalar multiplication, where r, v 1 , v 2 ∈ R,r ·( )v 1=v 2( )rv 1rv 2is a column array with two real entries, and so is in R 2 . Next, this checks (7).( ) ( ) ( ) ( ) ( )v 1 (r + s)v 1 rv 1 + sv 1 v 1 v 1(r + s) · === r · + s ·v 2 (r + s)v 2 rv 2 + sv 2 v 2 v 2For (8), that scalar multiplication distributes from the left over vector addition,we have this.( ) ( ) ( ) ( ) ( ) ( )v 1 w 1 r(v 1 + w 1 ) rv 1 + rw 1 v 1 w 1r · ( + ) === r · + r ·v 2 w 2 r(v 2 + w 2 ) rv 2 + rw 2 v 2 w 2