Math Is Fun Forum

Corine

Guest

Linear algebra question

The following problem has me puzzled.  Any advice would be helpful!:

The 2 by 2 matrix below is called G.  I have to find every possible r such that G would be = to a both a shear and uniform dilatation.   

First row:  1, r
Second row:  3, 4

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Linear algebra question

How many points is this?
Are we in 2-D?
I am just trying to learn this.

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igloo myrtilles fourmis

Corine

Guest

Re: Linear algebra question

Yes, the matrix represents a linear mapping from R^2 to R^2.

Corine

Guest

Re: Linear algebra question

Oh and by the way, I wasn't sure how to create a matrix using LaTex, so I just separated the numbers in matrix using commas. 
First row implies the numbers in the first row of the matrix.
Second row implies the numbers in the second row of the matrix.
Just in case that wasn't clear...

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Linear algebra question

Look at the sticky post at the top of the help me forum.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear algebra question

What is the definition of shear and uniform dilation?  The definition of shear is different from the one I'm used to (as no matrix of that form could possibly be a shear matrix) and I can't seem to find uniform dilation anywhere except in partical physics.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Corine

Guest

Re: Linear algebra question

What is the definition of shear and uniform dilation?  The definition of shear is different from the one I'm used to (as no matrix of that form could possibly be a shear matrix) and I can't seem to find uniform dilation anywhere except in partical physics.

I found this website about shear matrices on Wikipedia (but am having trouble applying it to the question):
http://en.wikipedia.org/wiki/Shear_matrix

As for dilation (uniform):  It is a stretch in every direction by a single value.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear algebra question

If that's the definition of shear you're using (the only one that I think exists), then G can't possibly be a shear matrix.  The trace of such a matrix needs to be n (in your case, 2) but your trace is 1 + 4 = 5.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."