How Do s and t Function in Matrix Representations of Linear Systems?

[josh3189]

Need some help with questions like this. I'm quite confused as to how to answer it in general.

http://tinypic.com/view.php?pic=2rhu6f7&s=6

I've posted only the first first basic problem as I can do the rest if I understand the logic behind this.

I actually encountered this while I was studying the introduction to linear algebra system of linear equations chapter.

What do "s" and "t" represent in this?

 

[HallsofIvy]

4x+ 9y+ 7z= 14 is the equation of a plane in an xyz- coordinate system. Since a plane is two dimensional, it can be written as parametric equations in two parameters. s and t are the parameters. There is no single "correct" answer- there are an infinite number of ways to choose the parameters, giving an infinite number of answers. One very simple method is to solve for one of the coordinates, say, z, in terms of the others: z= 2- (4/7)x- (9/7)y and use x and y as parameters: x= s, y= t, z= 2- (4/7)s- (9/7)t.

 

[josh3189]

4x+ 9y+ 7z= 14 is the equation of a plane in an xyz- coordinate system. Since a plane is two dimensional, it can be written as parametric equations in two parameters. s and t are the parameters. There is no single "correct" answer- there are an infinite number of ways to choose the parameters, giving an infinite number of answers. One very simple method is to solve for one of the coordinates, say, z, in terms of the others: z= 2- (4/7)x- (9/7)y and use x and y as parameters: x= s, y= t, z= 2- (4/7)s- (9/7)t.

Yeah, I considered that but as you can see in the question s and t act as a constant for the whole column. If I consider s = x, it becomes a problem when I am solving it in terms of x. The question I posted basically has only one possible answer for those blanks or so my prof says. Any ideas?

 

[Ray Vickson]

Yeah, I considered that but as you can see in the question s and t act as a constant for the whole column. If I consider s = x, it becomes a problem when I am solving it in terms of x. The question I posted basically has only one possible answer for those blanks or so my prof says. Any ideas?

The sentence before the last one is incomprehensible: I have no idea what you are attempting to say. If your prof. claims there is only one possible form he/she is wrong. If we solve for x in terms of y and z we get
x = 7/2 -(9/4)*y -(7/4)*z, so if we set s = y and t = z we have
$$\pmatrix{x\\y\\z} = \pmatrix{7/2\\0\\0} + s \pmatrix{-9/4\\1\\0} + t \pmatrix{-7/4\\0\\1}.$$
However, if we solve for y in terms of x and z we have y = 14/9 -(4/9)*x -(7/9)*z, so if x = s and z = t we have
$$\pmatrix{x\\y\\z} = \pmatrix{0\\14/9\\0} + s \pmatrix{1\\-4/9\\0} + t \pmatrix{0\\-7/9\\1}.$$
Still other representations can be obtained.

Basically, you want to describe any point in the plane by a 2-dimensional coordinate system lying in the plane. That means giving an "origin"---any specific point in the plane-- and two "axes" lying in the plane and emanating from the origin. These would be any two linearly independent vectors <x,y,z> that solve 4x+ 9y+ 7z= 0.

RGV