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This is my HW (Linear algebra 2) and I need to find projection matrix, kernel and image of the projection.

V=$R^2$

So I have: subspace of V

$sp{(2,-3)}$

Than I found the projection matrix is:

$$ \begin{matrix} 4/13 & -6/13\\ -6/13 & 9/13 \\ \end{matrix} $$

but now I need to find kernel and image... I don't remember how to do that and I searched and google and I know I saw I need to find Ax=0 So is my kernel is 0?! I don't totally understand this

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    What does "$sp(2,-3)$" mean?2012-12-05
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    I would guess the space is $\mathbb{R}^2$ and $sp(2,-3)$ is the span of the vector $(2,-3)$. But the OP should answer that.2012-12-05
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    @JulianKuelshammer you are right... so sorry.. fixing2012-12-05
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    The determinant of your matrix is zero, so the kernel will not be trivial.2012-12-05
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    so is it always depend on the determinant?? if it was non zero? and what about the image?2012-12-05
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    You can find the kernel by solving the linear equation $Ax=0$. What the image is, is obvious if you recall what the projection map does, i.e. that it projects onto the given subspace. You should include some more information about what you already know.2012-12-05
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    @JulianKuelshammer don't know nothing more than I mentioned..2012-12-05
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    What is your definition of projection matrix? How did you come up with this projection matrix?2012-12-05
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    as we learned at class there is a pattern for projection matrix that is -> 1/(a^2+b^2)*{{a^2,ab},{ab,b^2}}2012-12-05

1 Answers 1

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The projection you constructed is precisely the orthogonal projection onto the span of $(2,-3)$. So the span of $(2,-3)$ is the image.

Being an orthogonal projection, its kernel is the orthogonal of its image, so the kernel is $V^\perp$, which you can write as the span of $(3,2)$.

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    So this will be the same for every projection matrix? like if I have to find for y=1/2x the image is also R^2 and the kernel is sp(-1,2)???2012-12-05
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    My bad, I didn't read the question carefully when typing. The image can never be all of $\mathbb R^2$, unless your projection is the identity matrix. I've edited the answer.2012-12-05
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    Thanks, so following my previous comment the image will be sp (2,1) and kernel as I said?2012-12-05
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    Yes.$\ \ \ \ \ \ \ $2012-12-05