Answer to Question #8026 in Linear Algebra for alauddin Qureshi

Question

Geometrically representation of basis of v=(3,4) when v1(1,0) v2(0,1) v=av1+bv2

Answer 1

v=av1+bv2
Let's denote v={xv,yv}
Then
xv=a*xv1+b*xv2
yv=a*yv1+b*yv2
Substitute corrdinates of v1,v2:
xv=a*1+b*0=3
yv=a*0+b*1=4
These two equations allow us to find a and b:
a*1+b*0=3 => a=3

a*0+b*1=4 => b=4
So we obtain that v=3*v1+4*v2