# Tagged Questions

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### Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}\mathbb C[X,Y]/I$. Obviously ...
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### What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
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### Find a projectivity to create a graph.

I have the tetrahedron {xyzt=0} in projective space with homogeneous coordinate (x,y,z,t). I need to create a graph but the tetrahedron in affine coordinate is {xyz=0} and I can't visualize the ...
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### For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
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Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
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### The general expression of plane through the intersection of other two planes

For two planes: $$A_{1}x+B_{1}y+C_{1}z+D_{1}=0$$ $$A_{2}x+B_{2}y+C_{2}z+D_{2}=0$$ Prove that any plane going through the intersection line of the previous planes could be expressed like where ...
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### Finding signatue of a symmetric matrix.

Is it possible to find the signature of a matrix without finding the eigenvalues of the matrix? I was hoping to use the Sylvester's Law of inertia but I don't remember any algorithm to diagonalize a ...
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### Question about geometry in a finite projective space

I apologize again for a dumb question! To add some context (though I think it'll largely be unnecessary): suppose $q$ is a prime, $F:= \displaystyle \mathbb{Z}/q\mathbb{Z}$ is a field. I've defined ...
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### Change of coordinates (referential system) mistake? Doesn't seem to yield the proper coordinates.

Let $\varepsilon$ be an affine space with referential system $R$ characterized by $O=(1,1,1)$ as origin and $B=(c_1,c_2,c_3)$ as its basis, which is the canonical. Now, lets define a new ...
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### Difference in surfaces described by equivalent quadratic forms

It is fairly straightforward to prove that a quadratic form $Q(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}$ can equivalently be written $Q(\mathbf{x})=\mathbf{x}^{T}M\mathbf{x}$ for ...
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### The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$

Let $X$ be irreducible algebraic set of projective n space. I am trying to show that: The set of all $k$-dimensional planes which intersects $X$ is closed in $G(k,n)$, where $G(k,n)$ is the ...
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### Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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### Transform one curve into another

I have been working on something for a while now, and I can't really get my head around it. I consider two curves with data points and want to determine the most optimal transform from one to another. ...
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### What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
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### What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
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### How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...