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Codimension of variety versus linear independence of defining homogeneous...
Let $p_1, ..., p_m \colon \mathbb{R}^n \to \mathbb{R}$ be $m < n$ linearly independent homogeneous polynomials each of degree exactly $d$, and assume they have nonzero intersection. Is the...
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Eisenbud commutative algebra corollary 10.7
The $\leq$ direction is clear to me, but I do not know how the other direction works: I understand all the steps but I do not understand how this proofs anything in terms of dim R.
View ArticleComputing the global and Hochschild dimensions of a free product of direct...
Let $k$ be a field (algebraically closed of characteristic 0, but I do not expect it to make a difference).Consider the algebra $A=k^{n+1}\ast k^{m+1}$, where $\ast$ denotes the free product of...
View ArticleQuestions about the proof of Theorem 13.4 in Matsumura‘s Commutative Ring Theory
I'm reading the Matsumura's Commutative Ring Theory and I'm trying to understand theorem 13.4.Suppose $A$ is Noetherian semilocal ring, $M$ is a finite $A$-module, $\mathfrak{m}$ is the Jacobson...
View ArticleDimension of $R\left/aR\right.$.
In my algebra course I was asked to solve the following problem:Let $R$ a finite type $K$-algebra and suppose $R$ is an integral domain. If $0\neq a\in R$ is not invertible show that $\dim...
View ArticleCompute $\dim_{\mathbb{Q}}(\mathbb{Q}[s^{\pm 1},t^{\pm...
In the paper the homological algebra of Artin groups by Craig C. Squire it is stated that the following ring:$$R=\mathbb{Z}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,(st+1)(s-1))$$seen as an abelian group has rank...
View ArticleIf sum of subspaces is $V$, show $V = U_1\oplus U_2\oplus\ldots\oplus U_r\iff...
Let $V$ be a vector space with $\dim V = n <\infty$ and $U_1, U_2,\ldots, U_r$ subspaces of $V$, whosesum space is all of $V$.Prove that$V = U_1\oplus U_2\oplus\ldots\oplus U_r\iff \dim U_1 + \dim...
View ArticleHow to compute the dimension of $V(f,g)$ in $\mathbb Z[x_1,...,x_n]$
Consider the ring $R = \mathbb Z[x_1,...,x_n]$ and let $\text{Spec}(R) = \{q \subset R : q \text{ a prime ideal}\}$ be the set of prime ideals of $R$. For $I \subset R$ an ideal, define $V(I) = \{q \in...
View ArticleWhy does different nondimensionalizations give different results? Although...
I have some problems with the non-dimensionalization of the Hamiltonian of motion in a Coulomb field.The Hamiltonian has a following form:$$H=-\frac{\hbar^2}{2\mu^*} \Delta_r-\frac{e^2}{\epsilon_0...
View ArticleHow many 2D objects fit into a 3D object?
Hoe many times can you stack 2D objects before it becomes 3D?I assume stacking 2-dimensional planes alone the 3rd dimension would never actually stack, as along the 3rd dimension, the 2-dimensional...
View ArticleSome exercises about vector spaces
Hey I want ot check if my solutions for this exercise are right. Can someone help me?Let $V$ be a finite dimensional $K$-vector space and $U_1, . . . , U_n$ a family of $K$-subspaces in $V$ .Show that...
View ArticleNon-zero counts in increasing dimensions
I am working on a presentation that shows the exponential increase as one increases the number of dimensions, and I'm trying to figure out a way to calculate all non-zero or null counts, which I'll...
View ArticleFind a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that...
I have got the following entrance exam question.Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are respectively spanned by...
View ArticleReferences for bounds on dimension of particular matrix spaces?
I'm looking for references and well-known results about bounds on dimension of particular matrix spaces.For instance, the first result that came up to my mind was a Flander's theorem which explains...
View ArticleWhat is the $\mathbb{F}_p$ dimension of $\mathbb{F}_p[G]$?
I heard recently that the $p$-rank of a finite abelian group (the number of cyclic components of size $p^n$) is given by $\dim_{\mathbb{F}_p}(\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G])$, which is a...
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Question about dimension of a vector space
I'm reading Exercises in Classical Ring Theory (T.Y.Lam) and there is a exercise:I'm not sure how to determine the left (right) dimension of the vector spaces (red underline in the above).My though is...
View ArticleFind subset of vectors which form basis
QuestionLet W be the subspace of $R^5$ spanned by$ u_1= (1, 2, –1, 3, 4)\\u_2 = (2, 4, –2, 6, 8) \\u_3 = (1, 3, 2, 2, 6)\\ u_4 = (1, 4, 5, 1, 8)\\u_5 = (2, 7, 3, 3, 9)$Find a subset of the vectors...
View ArticleIs $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$ true?
Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ refers...
View ArticleVC dimension of indicator functions is equal to pseudo dimension
I am reading the "Foundation of machine learning" by Mehryar Mohri (https://cs.nyu.edu/~mohri/mlbook/). In the proof of Theorem 11.8, it said the following statement, which I can not understand.Let $H$...
View ArticleWhat is the dimension and nature of this variety?
Let $1 < N \in \mathbb{N}$ and $x, a \in \mathbb{C}^N$ with $a$ fixed; also, let $b \in \mathbb{R}_{\ge 0}^N$ be fixed (this last bit can be weakened to the extent it makes no difference). For $n...
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