Parametric Representations and Parallelism in Linear Algebra Homework Statement

[zacman2400]

Homework Statement

suppose x=x0+tv and y=y0+sw are two parametric representations of the same line l in r^n
a. show that there are scalars t0 and s0 such that y0=x0+t0v and x0=y0+s0w
b. show that v and w are parallel

The Attempt at a Solution

a. same line thus
y0+sw=x0+tv
when s0w=t0w then y0=x0 (essentially the solution...I think)

b. not sure at all
I don't like saying because the problem states: parametric rep of same line then must be parallel, but this is the most obvious statement to me

 

[quantumdude]

For part b pick any two points on the line x_1 and x_2 corresponding to t_1 and t_2, respectively. Since the second parameterization is of the same line then there must exist s_1 and s_2 such that y_1=x_1 and y_2=x_2, respectively. Can you take it from there?

 

[zacman2400]

leading us to the idea that y1-x1=y2-x2 meaning 0=0, thus parallel

 

[quantumdude]

No, you need to slow down and be more careful. You're trying to show that w is parallel to v, which means that w is a nonzero scalar multiple of v.