# Questions tagged [dimension-theory-algebra]

For questions about notions of dimension, rank, or length used in abstract algebra (e.g. Krull dimension, homological dimensions, composition length, Goldie dimension). Questions about dimension of vector spaces, and rank of linear transformations are better placed under the [linear-algebra] tag.

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### Proving equality of column space to field

Question: Let $A$ be a matrix of $m\times n$ and $B$ matrix of $n\times m$ over field $F$. Given that $AB = I_m$ ($m\times m$ unit matrix), prove that the column space of $A$ is equal to $F^m$. My ...
0answers
34 views

### Derivative of Multidimensional Rosenbrock Function

im having trouble understanding the derivative of the multidimensional Rosenbrock Function as shown here: Derivative I understand how to derive multidimensional functions in generel like ...
2answers
42 views

### Let $T : U → V$ and $S : V → W$ be linear transformations. How is $\operatorname{rank}(ST)$ related to $\operatorname{rank}(T)$?

Let $T : U → V$ and $S : V → W$ be linear transformations. How is $\operatorname{rank}(ST)$ related to $\operatorname{rank}(T)$? I know that $\operatorname{img}(S \circ T) = \operatorname{img}(S)$. I ...
0answers
25 views

### Volume of a Hyper-sphere in n dimension

Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space. [Hint: The formulas are different for n even and n odd.]
1answer
31 views

### Feedback to Basis and Dimension of $A:= \{x\in\mathbb{R}^4 | x_2 + 3x_3= 0, x_1=x_2\}$

My question: Because of $\mathbb{R}^4$ I assume the vector has to have $x_1,x_2,x_3,x_4$ To calculate the basis I assume that $x_1= 1$ because $x_1=x_2 as$ the problem says and $x_4= 0$ because this ...
1answer
29 views

### If $l$ is an integer such that $k \le l \le m$, show that there exists subspace $X$ of $V$ such that $U \subseteq X \subseteq W$ and $\dim(X)=l$.

Let $U$ and $W$ be subspaces of a vector space $V$ such that $U \subseteq W, \dim(U)=k,$ dan $\dim(W)=m$ with $k \lt m$. If $l$ is an integer such that $k \le l \le m$, show that there exists subspace ...
0answers
54 views

### A question on a result.

I was messing around a little bit and I got his claim: Proposition: Let $X \subseteq A_n$, be an affine variety. Then $\dim X=\text{height }I(X)$, where $I(X)$ is the ideal generated by $X$. I know ...
0answers
57 views

### $X, Y$ subspaces of $V$, $\dim(X) \geq \dim(Y) \implies \exists f$ linear s.t. $f(X)=Y$

Suppose that $V$ is a $K$-vector space and $X, Y$ vector subspaces of $V$ with $\dim(X) \geq \dim(Y) \implies \exists$ a linear map $f : V \to V$ such that $f(X)=Y$. I do not know what strategy to ...
0answers
46 views

### Dimension counting in exact sequences of Hopf algebras

It is known that the category of commutative and cocommutative Hopf algebras over a field $k$ is an abelian category. So we can talk about exact sequences. Reading a paper I found the following ...
0answers
36 views

### Fibered product of varieties and dimension

I am trying to do exercise 3.1.5 in Liu's book which asks the following. Let $X,Y$ be algebraic varieties over a field $k$. Show that $\dim(X\times_k Y)=\dim X\, + \dim Y$. The hint is to use ...
1answer
60 views

### Does the number of elements in Vector space and the dimension of Vector space same? [closed]

The number of elements in any basis of a vector space is called the dimension of Vector space. In the Vector space F^n(F) and Null Space; the dimension of Vector Space and the number of Vectors in the ...
1answer
62 views

### For $0 \xrightarrow{f_0} \mathbb{C}^n \xrightarrow{f_1} \mathbb{C}^\ell \xrightarrow{f_2} \mathbb{C}^r \xrightarrow{f_3} 0$, what is $\ell$?

So the question is as follows: Let $0, \mathbb{{C}}^{n}, \mathbb{{C}}^{\ell}$ and $\mathbb{{C}}^r$ be $\mathbb{{C}}$-vector spaces (where $0$ is the trivial vector space), and let $f_i$, $(i=0,…,3)$, ...
0answers
103 views

### Prove that the sequence $(\dim{\ker(f^{k+1})}-\dim\ker(f^k))_k$ is decreasing?

Let $E$ a $\mathbb{K}$-space vector with $\dim(E)=n$. Let $f\in \mathcal{L}(E)$. I have to prove that the sequence $(\dim{\ker(f^{k+1})}-\dim\ker(f^k))_k$ is descreasing. I do not want to use quotient ...
2answers
271 views

### How many times harder is to think in 3D as compared with 2D?

I'm a chemist, currently going through a course about molecular symmetry and group theory applied to Chemistry. This subject is very demanding in terms of visualization in 3D space. To really grasp ...
1answer
405 views

### What is the “dimension” of a locally ringed space?

Let $(X,\mathscr{O}_X)$ be a locally ringed space. If it is a scheme, the natural notion of dimension is the dimension of the subjacent topological space (the size of the biggest chain of irreducible ...
0answers
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1answer
60 views

### Is the dimension of a Noetherian local ring equal to its associated graded ring?

For a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$, let $I$ be a primary ideal in $A$, the associated graded ring is $$\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$$
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1answer
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### If each number line also has the Imaginary number line in it, does that mean x, y, z is six dimensions?

Are we to assume that x is two dimensional? I can somewhat picture this, but I'm having trouble with a number line for i (square root of negative one) with more than one dimension. i is not a ...
1answer
55 views

### Dimension, graph of functions of several variable and it's visualization.

To visualise a scalar function of $n$ variables we consider its graph in $(n + 1)$ dimensional space. If $\mathit{f} :U \subset \mathbb{R}^n \to \mathbb{R}$ is a function of n variable its graph ...
1answer
317 views

### Using rank-nullity for rankA + rank(adj(A)) = n iff col(adj(A)) = nullA

Let A be an $n \times n$ matrix. Show that $A$ is not invertible and $\text{rank } A + \text{rank}(\text{adj}(A)) = n$ if and only if $\text{col}(\text{adj}(A)) = \text{null } A$. The rank-nullity ...
1answer
1k views

### Find the dimension of the subspace of R^4 spanned by the set {(1,0,0,0),(0,1,0,0),(1,2,0,1),(0,0,0,1)}. Hence find a basis for the subspace

GIven set is not Linearly dependent hence not a basis. So should we take basis as {(1.0.0.0),(0,1,0,0),(0,0,1,0),(0,0,0,1)} and give as dim(R^4) = 4 or any other solution is expected?
1answer
194 views

### Proof verification: determining the dimension of a polynomial ring from the going up theorem.

I decided to prove that for any field $k$, dim $k[x_1, \ldots, x_n] = n$. Every proof I've seen follows either of these two approaches: Noether normalisation (first prove that if $A$ is a finitely ...
1answer
173 views

### Hilbert-Samuel multiplicity of a local ring of positive dimension is positive?

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d>0$. Then $e(R)=(d-1)!\lim_{n\to \infty} \mu (\mathfrak m^n)/n^{d-1}$. (Here $e(R)$ denotes Hilbert-Samuel multiplicity of $R$) . ...
0answers
65 views

### Explicit computation of projective dimension

I want to study the projective dimension $h_p(M)$ of an $A$-module $M,$ which it was defined as the least integer $n\in\mathbb{N}$ ($0\in\mathbb{N}$) such that exists a projective resolution of length ...
0answers
80 views