Math Is Fun Forum

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Parametrized Lines and Matrices

Let 

be the line parametrized as

and let

be the plane with equation

(a) Prove that the matrix

maps all points on line

to points on plane

(b) Prove that the matrix

maps all points on plane

to points on line

Again, if you could show me how you solved it, that would be great. Thanks!

admin note: some of your Latex doesn't work on this forum so I have edited to get it displaying properly, Bob

Last edited by !nval!d_us3rnam3 (2019-06-25 05:28:04)

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"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Parametrized Lines and Matrices

hi !nval!d_us3rnam3

Not all of your Latex was showing correctly so I have removed the command that were causing this.  I think our server has less features than you.

For part (a) if you re-write the line as a column vector then you can multiply A x l.  Then check the value of x + y + z.

I've come up with a way to do (b) but it's not very elegant. Here goes:

Consider (1,1,-1) and (0,1,0).  These are two points in P and so may be considered as a suitable basis.

Multiply these by the second matrix, and show that both points lie on the line.  If the base vectors map onto the line then all points in the plane will also.

LATER EDIT:  Slight re-think.  Find A where the line crosses the plane.  Also pick two points in the plane, B and C.  Use AB and AC as the basis.  (Because these transformations are 'linear' it still comes out the same.)

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Re: Parametrized Lines and Matrices

I'm net entirely sure what you mean about part (a), can you walk me through how to do it?
Thanks!

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"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Parametrized Lines and Matrices

Now check x + y + z = 2 + 5t + 2 - 5t - 3 = 4 - 3 = 1

So the image point is on the plane.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

!nval!d_us3rnam3

Member

Registered: 2017-03-18

Posts: 46

Re: Parametrized Lines and Matrices

And as for part (b), what are "base vectors"? I'm still kinda confused.

EDIT: Nevermind, I think I got it. Thanks!

Last edited by !nval!d_us3rnam3 (2019-06-27 09:18:10)

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"If we wanna be great, we can't just sit on our hands" - 2017 NFL Bears draft

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Parametrized Lines and Matrices

I think for the second part, it is important to vectorize the equation and write out the dependence of z on the other two:
(x, y, 1-x-y)'

Thus if you multiply the transformation matrix to this vector

It will be:

x+y-1+x+y
3x+3y-1+x+y
5x+5y-1+x+y

=
2(x+y)-1
4(x+y)-1
6(x+y)-1

=
2p-1
4p-1
6p-1

=
t
2t+1
3t+2

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X'(y-Xβ)=0