0
votes
0answers
17 views

mean number of links in adjacency matrix

I have converted from an individual-level adjacency matrix to one for clusters and I am trying to show mathematically how I programmed up determining the mean number of inter-cluster links. I am not ...
12
votes
2answers
160 views

Is there any distinction between these products: scalar, dot, inner?

I hope you will forgive a math question that comes up in physics contexts where language is loose. This question migrated from Physics SE. I'm finding that I sometimes don't know what kind of product ...
0
votes
0answers
36 views

Is there a traditional name for the “eigenspace” function?

Let $A$ denote a field, $X$ denote an $A$-vector spaces, and suppose $\varphi : X \rightarrow X$ is a linear transformation. Is there a traditional name for the corresponding "eigenspace" function? By ...
1
vote
0answers
21 views

I need help understanding what r-th and s-th rows are.

Let E be the matrix obtained from the unit $n \times n$ matrix by multiplying the $r$-th row with a number $c$ and adding it to the $s$-th row, $r \neq s$. Let $A$ be an $n \neq n$ matrix. Then ...
1
vote
1answer
15 views

Terminology of “linear”, “quadratic”, etc. for multi-input functions

It is my understanding that, according to typical math terminology: The function $f(x, y) = x + y$ is "linear". Specifically, it's linear in both $x$ and $y$, but this is understood implicitly. ...
1
vote
1answer
41 views

Name for multiples of orthogonal matrices

Is there a name for a matrix which is a multiple of an orthogonal matrix? I.e. a square matrix $A$ which satisfies the condition $$A^TA = AA^T = \lambda I$$ where $\lambda$ is some scalar (which ...
0
votes
2answers
27 views

Line in vector form?

Given the line y=3x my book states it is $\left(\begin{array}{c}1 \\ 3\\\end{array}\right)$ as a matrix. Why is it not $\left(\begin{array}{c}3 \\ 1\\\end{array}\right)$, I thought the upper number ...
2
votes
1answer
39 views

What is a “component function” of a matrix that is a linear map?

The matrix is: $$ \pmatrix{ 1&2\\ 3&4} $$ and this matrix is an $\Bbb R^2 \to \Bbb R^2$ linear map. I am asked to explicitly write the component functions of $A$. What does "component ...
1
vote
2answers
37 views

What does the term “distinguished basis” mean?

I know what a basis is (talking about vector spaces here), but I don't know what a distinguished basis is. Can you please explain the difference to me? I did not grow up in an English-speaking ...
11
votes
0answers
272 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
37
votes
19answers
4k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
0
votes
1answer
38 views

In the context of vectors is there a difference between the terms “magnitude” and “length”?

I noticed vectors are usually said to have "length and direction" but then it is said that people want to find the "magnitude". Is this just a difference in terminology or is there something more to ...
2
votes
2answers
226 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...
2
votes
1answer
47 views

Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...
1
vote
0answers
22 views

A name for a linear map from primary to dual space

Is there a standard name for a linear function $f:\mathbb R^n\to(\mathbb R^n)^*$ defined on the standard basis $e_i, i=1,\dotsc, n,$ of $\mathbb R^n$ by $f(e_i)=e^i$, where $e^i, i=1,\dotsc,n,$ is the ...
2
votes
2answers
34 views

Is there an object which groups two vectors together?

Is there a single name for a pair of vectors that together describe a position and orientation? Like an "oriented point" or something like that?
0
votes
1answer
635 views

Minimal Spanning Set vs Basis of a vector space

I read the following in my textbook: Find as small a set of vectors that span the row space of $A$ as you can. Such a set is called a minimal spanning set. Is this terminology synonymous with ...
25
votes
2answers
927 views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
1
vote
1answer
78 views

What does it mean when a set is said to be a “finitely generated vector space”?

I've somehow managed to go through 3 years of my Maths degree without truly understanding what the term "finitely generated vector space" means. Generated by what? Generating what? And for what? I ...
4
votes
1answer
74 views

Is there a name for this “co-dual” vector space?

Let $V$ be a $K$-vector space, then the linear transformations $T:K\rightarrow V$ (where $K$ is considered a 1-dimensional vector space over itself) form a vector space $$ A = ...
1
vote
0answers
123 views

Support of a vector

What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.
0
votes
0answers
29 views

What's $\{x\in V|\exists p\in \mathbb{Z}^+ {\phi(T)}^p(x)=0\}$ called?

I'm studying rational canonical form right now. Let $\phi(X)$ be a irreducible monic polynomial in $F[X]$ and $T$ be a linear operator on a vector space $V$ over $F$, where $F$ is a field. Define ...
12
votes
2answers
597 views

What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines. What is ...
0
votes
2answers
93 views

Is LinearTransformation's matrix a square matrix?

Linear Transformation’s confusions. Is linear transformation's matrix a square matrix? Some books define linear transformation is $V\to W$, and say linear operator is $V\to V$. Some books define ...
0
votes
0answers
24 views

Terminology for multilinear functionals the sum of whose cyclic shifts is zero?

Let $\varphi$ be an $n$-linear functional on a vector space $X$. Suppose that $\varphi$ has the property that, for all $(x_1,\ldots,x_n) \in X^n$, we have $$ \varphi(x_1,\ldots,x_n) + ...
0
votes
1answer
66 views

What is a set function that returns another set of points called?

I have a set of points $S = \{x_i\}_{i = 1}^m, x_i \in \mathbb{R}^n \forall i$. Now, I have a set function $f$ which operates as follows: $$f(S) = GX^T$$ where $G \in \{0,1\}^{m\times m}$ and $X = ...
2
votes
1answer
84 views

What is a “nonzero equation”?

Does it mean that there is at least one non-zero coefficient? That the solution set does not include the zero vector? That the the equation is non-homogenous? I am asking because the phrase is used ...
0
votes
0answers
84 views

Unit adjoint eigenvectors

Q.What are unit adjoint eigenvectors? Below I give the context where I found the mathematical term 'adjoint eigenvectors': $$\vec{f_u}.\vec{e_s}=\vec{f_s}.\vec{e_u}=0$$ so that by resolving a ...
4
votes
3answers
1k views

What does “isomorphic” mean in linear algebra?

My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have ...
0
votes
0answers
32 views

Word for “linear operator whose singular values are 0 and 1”

An orthogonal projection maps an inner product space onto one of its subspaces, and has eigenvalues 0 and 1. I need to write about a more general operator, the composition of an orthogonal projection ...
5
votes
3answers
656 views

What is the difference between vector components and its coordinates?

Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...
13
votes
4answers
299 views

Why the words “inner” and “outer” to designate products?

Does anyone know what's the rationale for using the adjectives inner and outer for certain algebraic products? Also, I've seen the term exterior algebra. Does the exterior here have anything to do ...
5
votes
1answer
167 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
1
vote
2answers
721 views

What is an ordered basis?

To my understanding, $$e_1 = \{1,0,\ldots,0\},\quad e_2 = \{0,1,\ldots,0\}, \quad \ldots, \quad e_n = \{0,0,\ldots,1\}$$ is an ordered basis for a vector space of dimension $n$. But the group of ...
7
votes
2answers
848 views

solving linear system “by inspection”?

A text question is asking to solve some linear systems by inspection. My interpretation of "by inspection" is "by looking". For a linear system like $3x + 4y = 28$ $3x + 4y = 83$ you could say that ...
2
votes
1answer
41 views

What is the names of $A\vec{x}=\vec{b}$ linear equation system components?

Having $A\vec{x}=\vec{b}$ . What is the names of $A\vec{x}=\vec{b}$ linear equation system components?
2
votes
2answers
342 views

How to interpret “rank” of a matrix intuitively?

What is the physical interpretation of "rank" of a matrix ? Why is it called "rank" ?
0
votes
1answer
42 views

A “Linear” Mapping - What am I talking about?

My situation: I have a fixed initial state $|\psi_i \rangle$ which is a ($1 \times n$) column vector. I apply a linear operator $\hat{A}(\phi_{1,2,3,...,x})$, which has a number of variables, to ...
5
votes
2answers
132 views

Why the SVD is named so…

The SVD stands for Singular Value Decomposition. After decomposing a data matrix X using SVD, it results three matrices, two singular vactors U and V, and one singular value matrix whose diagonal ...
7
votes
6answers
22k views

What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
0
votes
0answers
93 views

What is the name for a non-square permutation matrix?

Consider a matrix that selects and permutes some but not all of the entries of a vector. That is a binary $n\times m$ matrix, where $n<m$, with a single one per row, for example ...
0
votes
2answers
65 views

What does “similar via the <insert matrix here>” mean?

If "$A$ is similar to $B$ via the matrix $P$, are we saying that, $P^{-1} A P = B $?
9
votes
2answers
386 views

Etymology of the word “normal” (perpendicular)

While the word "normal" is one of the most overloaded mathematical terms, in linear algebra, it is usually associated with the notion of being perpendicular to something, as in "normal vector" or ...
0
votes
1answer
23 views

Name of a maximum bound

I'm reading this paper, which uses the quantity $$\max_{x\neq0} \frac{x^T A x}{x^Tx}$$ where $A\in R^{n\times n}$ is nonsingular and $x\in R^n$. This quantity looks so familiar to me that I'm almost ...
0
votes
1answer
59 views

What is a “rotated” basis?

My text (p. 19) introduces the concept of a "rotated" basis without explanation. What properties or characteristics of a basis make it "rotated" with respect to another? What operation on one basis ...
2
votes
2answers
202 views

What are matrix coefficients in linear algebra?

What are matrix coefficients in linear algebra? And what does it mean "integer matrix coefficients"?
3
votes
1answer
1k views

Why are vector spaces sometimes called linear spaces?

I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more ...
2
votes
0answers
85 views

Simple linear operator?

From Wikipedia a linear operator T on a finite-dimensional vector space is semi-simple if every T-invariant subspace has a complementary T-invariant subspace. I wonder if there is a concept for ...
0
votes
1answer
129 views

What is a Jordan Cell?

Google has been surprisingly unhelpful for me. A homework problem from my algebra class asks me to Calculate p(A) where A is a Jordan cell and p is a polynomial. ...
9
votes
2answers
2k views

What is the difference between kernel and null space?

What is the difference, if any, between kernel and null space? I previously understood the kernel to be of a linear map and the null space to be of a matrix: i.e., for any linear map $f : V \to W$, ...