# Tagged Questions

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### finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
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### Prove is linearly independent

Prove that that the following subset $S \subseteq V$ in the respectively specified $K$- vector space $V$ is linearly independent a. $K=R$, $V=R[x]$, $S$= {$x^n-x^m| n,m ∈ R,$ n-even, m-odd}
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### When $ax+by+cz+d=0$ is a plane, $a^2 + b^2 + c^2 \neq 0$

I'm reading a book about equation of planes and an way to determinate the equation is to suppose a point $P = (x, y, z)$ And suppose also that $A=(x_0, y_0, z_0)$ is in the plane. $P$ is in the plane ...
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### Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
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### Vector spaces - $\min\{p\in\mathbb{N}|\text{ker}f^p=\text{ker}f^{p+1}\}=\min\{q\in\mathbb{N}|\text{im}f^q=\text{im}f^{q+1}\}$

$E$ is a $\mathbb{K}$ vector space, $f\in\mathcal{L}_\mathbb{K}(E)$. Let $p\in\mathbb{N}$ so that $\text{ker} f^p=\text{ker}f^{p+1}$ and $q\in\mathbb{N}$ so that $\text{im} f^q=\text{im}f^{q+1}$ ...
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### How to prove perpendicular vectors problem

If I have: $$|\underline a + \underline b| = |\underline a - \underline b|$$ how do I prove that $\underline a$ is perpendicular to $\underline b$?
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### Linear Algebra, Vector Space: how to find intersection of two subspaces ?

$${ W = Sp\{{(1,3,4),(2,5,1)\}}\\ U = Sp\{{(1,1,2),(2,2,1)}} \}$$ Find a span $${U \bigcap W}$$ First time using Math latex, pretty hard.
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### finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated. a) If $S$ is a set ...
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