# Tag Info

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Here are some more examples: $C[a,b]$, the set of continuous real-valued functions on an interval $[a,b]$. This abstract vector space has some very nice properties that make it very good for a first-semester linear algebra course: a. It has a natural inner product on it, given by $\langle f, g \rangle = \int_a^b f(t)g(t) \, dt$ b. It contains the (infinite-...

7

I wouldn’t feel bad about leaving it out, but I think it’s a valuable conceptual example for understanding matrix algebra. Computing the QR decomposition is equivalent to applying Gram-Schmidt orthogonalization to the columns, and I think it’s really instructive to see how this corresponds exactly to the fact that Q is orthogonal and R is upper triangular (...

6

The vector space $V = C^{\infty}(\mathbb{R},\mathbb{R})/\mathbb{R}[x]$ of smooth functions modulo polynomials. Note that $d/dx \colon V\to V$ is an isomorphism, so that we have a nice inverse $\int \colon V \to V$, taking the class of a function to the class of an antiderivative. So suddenly, the indefinite integral operation is well-defined. Note that it ...

5

The spin states of an electron form a two-dimensional vector space over the complex numbers. Designate "spin up" and "spin down" for a basis. The vector space structure is a consequence of the linearity of the Schrodinger equation. The computer science slant on this situation uses the word "qubits."

5

Let $\Omega$ be a set, and let $\mathcal A$ be an algebra of subsets of $\Omega$. Then $\mathcal A$ is a vector space over the field $\mathbb F_2 = \{0,1\}$, with the operation $$E \Delta F = (E \cup F)\setminus (E \cap F)$$ as addition, and $$0E = \varnothing,\qquad 1E=E$$ as scalar multiple. Any finite-dimensional vector space over $\mathbb F_2$ has ...

5

Solving least squares problems by QR factorization is much more numerically stable than solving them by Cholesky factorization of the normal equations. This can easily be demonstrated on an ill-conditioned test problem.

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Have these students had trigonometry? If so, they may have seen the formulas for rotating a point around the unit circle. $$x' = x\cos(\theta) - y\sin(\theta)$$ $$y' = x\sin(\theta) + y\cos(\theta)$$ If we think of the slope $m=\frac{y}{x}$ relative to an origin of $\langle0,0\rangle$, then $x$ and $y$ can be thought of as just a point that can be ...

5

Here's the algebra-based proof I've used in a college algebra class. Perpendicular lines are defined as meeting at a right angle. Assume that we know the Pythagorean and distance formulas. A possible lemma is that slope of a line indicates how much $y$ increases for a 1-unit increase in $x$ on that line. Given that $m = \Delta y / \Delta x$, when $\Delta x = ... 4 Since you used "$90^\circ$", let me suggest something motivated by trigonometry. Define the relative-slope between two lines by $$m_{rel}=\frac{m_2-m_1}{1+m_2m_1},$$ from the trigonometric identity $$\tan(\theta_2-\theta_1)=\frac{\tan\theta_2-\tan\theta_1}{1+\tan\theta_2\tan\theta_1}.$$ If the lines are parallel [i.e.$\theta_2-\theta_1=0^\circ$],... 4 The following comes from https://projecteuclid.org/download/pdf_1/euclid.afm/1485893376 Let$(V,|\cdot|)$be a normed vector space. Define$v \perp w$if$|v-w|^2 = |v|^2+|w|^2$. Define a "normed perpendicularity space" as a normed vector space where the set of vectors orthogonal to a given vector is always a subspace. Then$(V,|\cdot|)$arises ... 4 The set of solutions to a system of linear homogeneous ODEs is a vector space, and the dimension of this vector space is equal to the total order of the system. The idea that every solution is the linear combination of some "special" set of solutions is just a statement that we can always find a basis of$n$elements in an$n$-dimensional vector ... 4 The set of basis vectors is just called the basis in English, and so your topic is usually called change of basis. Google is telling me that "repère" is translated as "coordinates", which would be the family of scalar variables that are used to define each point in the space in terms of the basis vectors. Bonne chance! 3 Some more (rather general) examples of vector spaces over any field$\mathbb{F}$: For any set$S$, the space$\mathbb{F}^S$of mappings from$S$to$\mathbb{F}$, with pointwise addition and scalar multiplication. (Actually, every vector space is isomorphic to a subspace of such a space, due to the existence of a basis.) More generally, if$S$is a set and$...

3

Below are two elementary/expository papers that I know for generating strange examples, although how interesting they are will vary from person to person (however, a connection with the special theory of relativity does exist), and unless some additional conditions are imposed I don't believe they're finite-dimensional. Nonetheless, I think it's worth giving ...

3

Elie Cartan wrote a paper in 1935 that was foundational for differential geometry. It is titled La Méthode de Repère Mobile, La Théorie des Groupes Continus, et Les Espaces Généralisés. In English, we often refer to the "method of moving frames." Here is a link to a review of the paper

3

Let us define two lines $L_1: y=m_1x+b_1$ and $L_2: y = m_2x+b_2$ to be perpendicular if their intersection exists and forms a right angle. Clearly $m_1 \neq m_2$. Let $P=(x_o,y_o)$ be the point of intersection. Then, $$m_1x_o+b_1 = y_o = m_2x_o+b_2$$ Observe $b_2-b_1 = (m_1-m_2)x_o$ this will be important later. Furthermore, select $x_2 > x_o$ and ...

3

In the US, Cayley-Hamilton isn't typically taught in a first course in Linear Algebra (non proof-based, intended for students majoring in the sciences and engineering), but is typically taught in a second more advanced course for mathematics majors. You can see this by looking at textbooks intended for these two different kinds of linear algebra courses. ...

3

Here are projects I have written and used. I assign these to student groups of 3-6 students, mostly Engineering majors, mostly sophomores. In class time is about 30 minutes per project to introduce the set up; then they work out of class for about 2 weeks per project (and come to office hours frequently during that time). Linear programming Camera matrices ...

2

This may be too advanced for some students in a first-semester linear algebra course, but those who have had some Physics may be impressed by the diagonalization of a moment of inertia tensor. Take some irregularly-shaped 3-dimensional object, and form a $3 \times 3$ matrix describing all of the components of the moments of inertia around various axes; the ...

2

Not a software exactly, but super useful for those in the process of learning row reduction is the website: Linear Algebra Toolkit, by P. Bogacki. You just choose the number of rows and columns and it will produce the reduced row echelon form, and show each step along the way with the indicated row-op clearly stated. I've used this for years. For example:

2

We can think of these transformations from two perspectives: as mappings of the plane or as graphs of pre/post composition of the function. For instance, the map $T: \mathbb{R}^2 \to \mathbb{R}^2$ given by $(a,b) \mapsto (a+2,b)$ is the translation of the plane two units to the right. Let $\phi: \mathbb{R} \to \mathbb{R}$ be the function $\phi(x) = x-2$. ...

2

Here is an approach which works only(?) in two dimensional case and seems reasonably satisfying to me. Hope it helps to clarify the question! Let $V$ be a two-dimensional real vector space. Consider a structure on $V$ consisting of: A linear transformation $r$ on $V$ such that $r^2 = -1$ chosen up to sign. A nondegenerate skew-symmetric bilinear form $s$ on ...

1

Yes, (ignoring diacritical marks) Cartan's "repere mobile" is "moving frame", but that refers to a more complicated idea than I think will come up in basic linear algebra. In basic linear algebra, it would make sense to just say "frame", rather than "moving frame", but/and the plain "frame" is better ...

1

This is the example that made me realize the importance of abstract linear algebra as an undergraduate: the edge spaces and the vertex space of a graph (https://en.wikipedia.org/wiki/Edge_and_vertex_spaces). The edge graph of a graph $(V, E)$ is the vector space of functions from $E$ to the 2-element field. The vertex space of a graph is defined likewise. ...

1

Expanding on the answer by James Cook, the whole zoo of spaces in poly-linear algebra: the dual space $V^\star$ of a given space; the space $V^\star\otimes V^\star$ of all bilinear forms; the space $V^\star\otimes W$ of all linear operators from $V$ to $W$; for $V=W$, the subspaces of symmetric and anti-symmetric operators; the spaces $\Lambda^k(V)$ of anti-...

1

I'll give a less elegant answer: because the dilations and translations are easy to understand and implement. If we were interested in preserving shape under the transformations then surely rigid motions or orthogonal transformations would be a better scope for the discussion. Sadly, highschool students (in the USA) do not typically learn linear algebra, ...

1

In general, manifolds have not a norm, but a metric, which is a function that takes two points as input and gives their distance as output. A space with addition/subtraction and a norm has a metric (the distance between two points is the length of their difference), and metric that respects linearity is a norm (the length of a vector is the distance between ...

1

The text, "Linear Algebra Done Right" Springer(1997) by Sheldon Axler explicitly addressed the same problem. Matrices first appear in the last chapter (10).

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It seems to me that a discussion of how to make any function $f: A \rightarrow B$ into a bijection might be in order. First, we can deal with onto by replacing $B$ with $f(A)$. So, let $g: A \rightarrow f(A)$ be the function given by $g(x)=f(x)$ for each $x \in A$. Next, we may need to make the domain smaller, for each non-empty fiber f^{-1} \{ b \} = \{ a ...

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