# Questions tagged [linear-algebra]

For questions about linear algebra, a mathematical field studying vector spaces and matrices.

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### A modern reinterpretation of Vandermonde's mathematical achievements

I already posted a question on the Mathematics Stack Exchange about a related topic, but after posting it, I realized that it is more appropriate for this site, so I am posting it again in this site. ...
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### Why did Vandermonde become interested in algebraic problems?

I recently became interested in a person named Vandermonde when I posted a question on this site about the history of determinants and read the answers and comments. He was a person who loved music ...
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### Origins of the use of matrices for the calculus of binary relations

Nowadays it's common to represent a binary relation $r$ (a subset of the Cartesian product of two sets $A$ and $B$) as a 0,1-matrix whose rows correspond to elements of $A$, whose columns correspond ...
• 637
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### How was Laplace able to propose the Laplace expansion?

I read Sylvester's 1850 paper, which is on file here. If you look at page 147 of this file, you will see the following sentence: Imagine any determinant set out under the form of a square array of ...
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### History of Invertible linear transformations

Who studied invertible linear transformations for the first time? I would guess these linear transformations weren't studied for the sake of studying invertible linear transformations, they probably ...
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### Origin of decomposition theorem

There are many different types of decomposition theorem in linear algebra. For example, there are primary decomposition theorem, cyclic decomposition theorem, etc. But I became curious about the ...
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1 vote
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### Origin of Multi-linear map

Multi-linear map is used to define the determinant. However, since the multi-linear map itself does not have linearity, I feel a sense of heterogeneity when I compare this concept with the contents ...
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### Why is Einstein summation named after Einstein?

Why is Einstein summation of tensors (summation of repeated indices) named after Einstein? "Einstein rule" in the Encyclopedia of Mathematics only says: This rule was proposed by A. ...
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### Earliest real-world uses of calculus and linear algebra

I want to illustrate in class that real-world applications of mathematics might take time to come to fruit. In this context, I want to find what the earliest real-world applications of Calculus and ...
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### Who discovered that the Lanczos method can only calculate extremal eigenvalues of large matrices?

The Lanczos tri-diagonalization process is widely or even routinely used today. It is said that it is useful for obtaining the extremal eigenvalues, but useless for medium eigenvalues. But who ...
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### What can I read to learn the history of multivariable calculus?

People have been doing calculus of several variables since well before the concepts of vectors, matrices, and linear algebra were formalized. Where can I learn about the development of multivariable ...
75 views

### Sparse matrix ("matrice creuse") etymology in French

I am looking for the etymology of matrice creuse. According to Wikipedia, it seems James Joseph Sylvester used the term "matrix" in 1850, and Harry Markowitz used the term "sparse ...
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### What was the role of Schmidt in derivation of the Gram-Schmidt process?

When reading the section related to Gram-Schmidt process in the book Linear Algebra and Its Applications by Gilbert Strang, I found a foot note that says: If Gram thought of it first, what was left ...
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### Negative coefficients in the barycentric calculus

The barycentric calculus of Möbius involves formal sums of expressions of the form $mP$ where $m$ is a real number and $P$ is a point, where $mP$ is to be thought of as $m$ units of mass located at $P$...
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### Understanding Grassmann's Approach to Algebras over Vector Spaces?

According to Hermann Grassmann and the creation of linear algebra by Desmond Sander, Grassmann was able to identify all the important notions in linear algebra in his book "Ausdehnungslehre"....
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### Earliest mention of permutation matrices, or equivalent? More generally, matrices for arbitrary functions between finite sets?

Permutation matrices I assume have a long history, and would be surprised if they were first considered only long after the work of Shur just after 1900, on the representation theory of $S_n$. ...
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### Sylvester's Quote on Determinants

What does the following quote by Sylvester mean? "A general algebraical determinant in its developed form may be likened to a mixture of liquids seemingly homogeneous, but which, being of ...
99 views

### Where does the "operator to the right" notation originate?

If any of you have ever written code in DirectX, you're sure to have noticed that applying a linear operator $A$ to a vector $x$ is done as $xA$, instead of the (nowadays usual) $Ax$. I wanted to know ...
159 views

### Why did Clifford Algebra suddenly gain a resurgence under the name of Geometric Algebra in recent years?

To my understanding, this new Geometric algebra thing is actually nothing more than years old Clifford Algebra. Yet it is advertised by many of its proponents as a fundamentally new thing; why is that?...
201 views

### How and by whom was the concept of generalized eigenvectors developed?

In general, a linear operator on a complex vector space need not always have an eigenspace decomposition. But it will always have enough generalized eigenvectors to provide a decomposition of the ...
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### Where did block matrix multiplication appear?

I am curious about who first noticed that block matrices can be multiplied blockwise. There is a section about matrices partitioned into submatrices that describes block matrix multiplication in "An ...