Question

Describe the object in R^4 that is represented by the vector equation and find parametric equations for it? for example like (x_1, x_2, x_3, x_4) =(-1,0,4,2) + t_1(-3,5,- 7,4) + t_2(6,3,-1,2)

Answer 1

Like how do you know when its a line and when its a plane

Answer 2

A line has one dimension, a plane has two dimensions, anything greater is a hyperplane.

Answer 3

How bout in terms of solutions?

Answer 4

What do you mean by that? Solutions to a system of linear equations?

Answer 5

Yeah I asked my TA and he was like if it's one solution then it's a line or something?

Answer 6

If there is a unique solution it is a single point (a single vector).

Answer 7

okay I think dimensions make more sense, how would you check what the dimensions are for that general equation

Answer 8

(x_1, x_2, x_3, x_4) =(-1,0,4,2) + t_1(-3,5,- 7,4) + t_2(6,3,-1,2)

Answer 9

is it two dimensions since there are t_1 and t_2

Answer 10

In that case the solution is a plane

Answer 11

how did you know? from the t_1 and t_1 or the 4 coordinates?

Answer 12

Indeed because there are two parameters.

Answer 13

so if there are t_1 and t_2 and t_3 then there it is a hyperplane

Answer 14

yeah

Answer 15

Oh okay thank you!