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October 16th, 2013, 07:47 PM   #1
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Finding the matrix in the standard basis for the projection

The problem is to find the matrix in the standard basis for the projection R^2 of the projection onto the line 2y=x.

So this means the solution should be a linearly independent set of 2 vectors, but how do I figure this out, is it just a matter of picking arbitrary points for x and y?
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October 17th, 2013, 05:43 AM   #2
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Re: Finding the matrix in the standard basis for the project

Are you sure the question isn't to find the matrix for reflection in the line 2y = x?

In any case, are you not looking for a matrix rather than a basis?
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October 17th, 2013, 11:10 AM   #3
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Re: Finding the matrix in the standard basis for the project

A line is a one dimensional subspace. Choose a nonzero element of the line, for example (2,1). If the length of this vector is not 1 then divide it by its length so you get a unit vector t=(2/sqrt(3),1/sqrt(3)). For the projection you should decomposite every r vector into a vector parallel to the line and a vector orthogonal to the line. In fact the projection means that you maps the r to the parallel component of it. You can express it with the introduced t unit tangent vector as:
r' = (tr)t.
Indeed, the scalar product in the parenthesis is |t||r|cos(theta)=r cos(theta) that is the length of parallel component. Multiplying t by this length is the projected vector r'.

Now you can conclude that the projection is a linear transformation, because scalar product is linear. You can also read the element of the matrix if you reexpress it with (x,y). The formula says:
x' = (t1*x+t2*y)*t1 = (t1*t1)x + (t1*t2)y, so the first row of the matrix is [t1*t1 t1*t2].

Note: The method works in higher dimension, as well. Sometimes the transformation of one dimension projection is written as

that is

(Sadly I forget the name of this special production.)
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October 22nd, 2013, 08:01 AM   #4
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Re: Finding the matrix in the standard basis for the project

Thank you very much for the help. I wa wondering, if the matrix firme by the projection in part a by a 90 degree counter clockwise rotation would it just negate the values in the x column?
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October 22nd, 2013, 05:21 PM   #5
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Re: Finding the matrix in the standard basis for the project

Quote:
Originally Posted by 84grandmarquis
Thank you very much for the help. I wa wondering, if the matrix firme by the projection in part a by a 90 degree counter clockwise rotation would it just negate the values in the x column?
You are using words in very strange ways that make me wonder if you are clear on their definitions. You cannot talk about a "projection by a 90 degree counter clockwise rotation". A projection is not a "rotation". And a "projection onto a line" does NOT give "a linearly independent set of 2 vectors" because a line is one dimensional.

I can't help but wonder if you are not using "projection" where you should be using "matrix".
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