Suppose U is a subspace of V such that V/U is finite dimensional. Can we say...
Suppose U is a subspace of V such that V/U is finite dimensional. V/U is the quotient sapce, namely the set of all affine subsets of V parallel to U.I think we cannot show that V is finite dimensional,...
View ArticleRank and dimension of matrix in a linear map
Let's say I map a $3 \times 1$ vector $\underline v=(x, y, z)$ by multiplying it with a $3 \times 3$ matrix of rank $2$. Would I be correct in thinking that it transforms all points in 3D space into a...
View ArticleAbout the regular representation of weak hopf algebra
In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now consider...
View ArticleDimension of a Matrix subspace
What is the dimension and the number of basis vectors for a subspace of 3×3 symmetric matrices?Earlier my professor told us that the dimension and the number of basis vectors for a subspace are the...
View ArticleA proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't...
View ArticleIs the dimension of this subspace 1?
"Let $V=M_{2\times2}(\mathbb{R})$ denote the vectors space of all $2\times2$ matrices with real number entries. Determine which of the following subsets are subspaces of $V$. If it is a subspace, find...
View ArticleDimension of $\mathcal{O}_X(X-\{P\})$.
Let $X$ be a smooth projective and irreducible curve and $P\in X$. I am asked to show that the dimension (as a $k$-vector space where $k$ is a algebraically closed field) of $\mathcal{O}_X(X-P)$ is...
View ArticleClassical Krull Dimension of Commutative Rings
I've been looking at the extension of Krull dimension to non-commutative rings as defined, for example, in On the Krull-Dimension of Left Noetherian Left Matlis-Rings [Krause, Mathematische Zeitschrift...
View ArticleTensor product of simple $sl_2$ modules
I am working on the following problem: Let $M(n)$ be the finite-dimensional, simple $\mathfrak{sl}_2(\mathbb{C})$-module with highest weight $n\in\mathbb{N}_0$. Show that the module...
View ArticleIs the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq...
If V is the linear space over the field R of real numbers, and $U_1, U_2, U_3$ are subspaces of this space. Is the inequality always true: $\dim_R (U_1 + U_2 + U_3)\leq \dim_RU_1+\dim_RU_2+\dim_RU_3$I...
View ArticleTo calculate the dimension of a vector space
Let $E$ and $F$ be two subspaces of $\mathbb{R}^n$, and let $$G = \{\begin{pmatrix} X \\ Y \end{pmatrix}\in \mathbb R^{2n} \mid X+Y \in E, Y \in F\}$$. I am trying to calculate the dimension of $G$,...
View ArticleCodimension 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...
View ArticleInequality of module rank using annihilator
I am reading a paper and am somewhat confused by what I think should be a pretty elementary step, so any help would be really appreciated! Here's the question:Let $R$ be a noetherian ring with...
View ArticleRank preservation of inverse limit of modules
I am asking about the properties of rank over an inverse limit.Suppose we have a noetherian ring $R$ with uniformizer $\pi$ and noetherian ring of formal power series $R[[X]]$. Let $\mathscr{R}$ be the...
View ArticleProof verification: dimension of inverse limit of modules
I'm looking for clarification as to whether or not the following proof is valid. I am unsure in justifying the last paragraph.Thank in advance as always,MClaim: Let $R$ be a ring and $(M_{k} ,...
View Articlemisunderstanding on real algebraic varieties
Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find easy...
View ArticleReal-valued dimension
Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$.Does there exist an example of the following?A commutative ring with unity $R$A mapping $\operatorname{d}:...
View ArticleComputing the height of an ideal...?
I hope I'm not overbearing in this site. Yes, I'm still struggling.If you can, I have a question about primary decomposition that still needs help, you can find it in my page.Now I wanted to find the...
View ArticleHow to prove the isomorphism $ \mathcal{O}_{X,c} \cong \mathcal{O}_{V,c} /...
If $V$ is an affine variety and $X, Y \subsetneq V$ are pure dimensional and closed, and $c$ is a zero dimensional irreducible component of $X \cap Y$.Then how to prove $ \mathcal{O}_{X,c} \cong...
View ArticleProof of Proposition 11.20 of Atiyah-Macdonald
I struggle with verifying the pole order inequality asserted in the proof of proposition 11.20. (Full statement and proof of the proposition can be found here: Atiyah-Macdonald 11.20 and 11.21)My...
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