Mat233 lecture 9 linear combinations & spans. [2] the set of all linear combination is denoted by. And called the subspace spanned (or generated) by (cid:1874)(cid:3041) (cid:1874) . Hence (cid:1875) {(cid:1874)(cid:3041) (cid:1874) } (cid:3643) (cid:1875) =(cid:1874)(cid:3041) + (cid:1874) for some (cid:3041) . Let p = 2, and n = 2 (i. e. we are in 2 dimensions) We can add them (we need to find c1 and c2 so that this is true. Then apply the reduction algorithm to find the solution (do this on your own and try to get this answer we have here: To solve this, we write the augmented matrix: (cid:2869)+(cid:884)(cid:2870)=(cid:886) (cid:884)(cid:2869)+(cid:885)(cid:2870)=(cid:887) [(cid:883) (cid:882) (cid:2779) (cid:2780)] (cid:882) (cid:883) (cid:3642){(cid:2869)= (cid:884) (cid:2870)=(cid:885: this means that the vector (cid:1875) is contained in the span: (cid:4672)(cid:886)(cid:887)(cid:4673) {(cid:1874) (cid:2869),(cid:1874) (cid:2870)} (cid:1874) (cid:3041) (cid:1874) (cid:3041) This leads to a new linear system which is an equivalent system. We can therefore conclude that (-2, 3) solves (s)
