Consider aX + bY = 0
Either a Y exists st aX + bY = 0 only if b=0, in which case X and Y are Linearly Independent, or Y = pX for all Y and M is at least one-dimensional.
If X and Y are LI, consider aX + bY + cZ = 0.
Either a Z exists st aX + bY +cZ = 0 only if c=0, in which case X, Y and Z are LI, or Z = pX +qY for all Z and M is at least two-dimensional.
Eventually you reach a point where no W exists st aX + bY + cZ +....+ dU + eW = 0 only if e = 0, and then every W can be expressed as a linear combination of X, Y, Z,....,U which is a basis of M of dimension r (number of LI members X,Y,Z....,U).
"Vector" was never mentioned.