Linear algerbra Let u 203 and v 419 find two vectors in

Linear algerbra

Let u= < 2,0,3 > and v= < -4,1,-9 > find two vectors in span {u,v} that are not multiples of u or v and show the weights on u and v used to generate them.

Solution

u= < 2,0,3 >

v = < -4,1,-9 >

any vector x = au +bv will be in span {u,v}

< -2,1,-6> & <6,-1,12> are in span {u,v}

clearly they are not multiples of u or v ,

< -2,1,-6> = < 2,0,3 > + < -4,1,-9 > = u +v ,so a = 1,b = 1

<6,-1,12> =  < 2,0,3 > - < -4,1,-9 > = u - v ,so a = 1,b = -1

there are infinitely many vectors which are in span {u,v} that are not multiples of u or v .