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When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.

Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which "feel" very different from $\mathbb{R}^n$. It can be a little difficult to justify the extra work involved in making the abstraction because of this.

I have a short list of examples which I like to use:

  1. The space of polynomials of degree less than or equal to $n$ for some $n \in \mathbb{N}$. Interesting applications include interpolation and splines.
  2. The space of rational functions with a given denominator. An interesting application is partial fraction decomposition.
  3. The space of $n \times m$ real matrices. One application here is to image analysis, if you view the entries of the matrix as indicating brightness of a pixel located on a screen. Projecting into a lower dimensional subspace is a form of image compression. One super interesting application is mind reading.
  4. The space of sequences satisfying a linear recurrence relation. The shift operator is a linear operator from this space to itself. Finding the eigenbasis of the shift operator allows one to extract an explicit formula for the $n^{th}$ term of such a sequence. One notable example is an explicit formula for the Fibonacci sequence.
  5. Real Homology and cohomology of a simplicial complex. The vector spaces are the spaces of chains/cochains and the linear maps are the boundary maps. I have never used these as examples in an intro linear algebra course, but it seems like one could assign a student project about this with significant scaffolding.

I am hoping to generate a "big list" which would be useful to teachers of linear algebra.

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    $\begingroup$ I would add, solution set for $n$-th order ODE. Or, solution set of $n$-first order ODEs. Nonhomogeneous cases give interesting examples to illustrate the quotient space concept. Also, the space $L(V,W)$ of linear transformations $V$ to $W$ where $V,W$ are finite dimensional vector spaces. Like the solution set example, $L(V,W)$ is itself a finite dimensional subset of the space of all functions from $V$ to $W$. We can also look at multivariate polynomials as a finite dimensional set of functions with which approximations for multivariate functions ala Taylor are found. $\endgroup$ Commented Jun 16, 2021 at 15:13
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    $\begingroup$ A field is a vector space over itself, which is a nice example for a finite vector space. $\endgroup$
    – FormerMath
    Commented Jun 16, 2021 at 17:03
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    $\begingroup$ I would appreciate an explanation of the downvote! $\endgroup$ Commented Jun 16, 2021 at 18:00
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    $\begingroup$ I would appreciate an explanation of the downvote! --- Me too, for that matter. On the other hand, seeing your comment led me to realize that I forgot to upvote your question! (I've been working on day-job stuff the past hour and just came back for a short break.) $\endgroup$ Commented Jun 16, 2021 at 18:57
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    $\begingroup$ Surely you mean the space of polynomials of degree less than a given degree, otherwise it isn't a vector space (it wouldn't be closed under subtraction) $\endgroup$
    – Evariste
    Commented Jun 17, 2021 at 6:25

14 Answers 14

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Here are some more examples:

  1. $C[a,b]$, 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-dimensional) vector space $\mathcal P$ of all polynomial functions (restricted to $[a,b]$) as a subspace, as well as every (finite-dimensional) vector space $\mathcal P_n$ of polynomial functions of degree $\le n$ for given $n$.

    c. Considered as an inner product space with respect to the inner product in (a), orthogonal projection onto $\mathcal P_n$ produces the best possible polynomial approximation of degree $n$ on $[a,b]$.

    d. It also contains the (infinite-dimensional) vector space $\mathcal F$ consisting of all convergent Fourier series on $[a,b]$, as well as every (finite-dimensional) vector space $\mathcal F_n$ of Fourier series of order $\le n$.

    e. Considered as an inner product space with respect to the inner product in (a), orthogonal projection onto $\mathcal F_n$ produces the best possible approximation as a Fourier series of order $n$ on $[a,b]$.

  2. Subspaces of the examples in the OP:

    a. The vector space of upper-triangular $n\times n$ matrices

    b. The vector space of symmetric (or skew-symmetric) $n \times n$ matrices

    c. The vector space $\mathcal P_{\textrm{even}}$ of polynomials with only even-degree terms, and the corresponding vector space $\mathcal P_{\textrm{odd}}$. Notice that $\mathcal P = \mathcal P_{\textrm{even}} + P_{\textrm{odd}}$, a nice example of subspace addition, and moreover the differentiation operator $D:\mathcal P \to \mathcal P$ is a linear transformation that maps these two subspaces onto each other.

  3. Vector spaces of functions of a particular form. These can be somewhat artificial but nevertheless have nice pedagogical properties. For example, for any fixed real numbers $k, r$, the set $V$ consisting of functions of the form $f(x) = A\sin kx + B\cos kx + Ce^{rx}$ is a subspace of $C[a,b]$. The differentiation operator $D$ maps $V \to V$, and if you choose the natural basis for $V$, then the matrix of $D$ relative to that basis has a nice geometric interpretation as a $90^\circ$ rotation in the plane spanned by $\sin kx$ and $\cos kx$, followed by a dilation by a factor $k$ in that plane and a dilation by a factor $r$ in the direction of the subspace spanned by $e^{rx}$.

  4. Finally, let's not forget: $\mathbb C$, considered as a 2-dimensional real vector space. The isomorphism with $\mathbb R^2$ is so obvious that sometimes we forget these are not "actually" the same set. (Yes, I know $\mathbb C$ is sometimes defined as $\mathbb R^2$ with additional structure, but it need not be defined that way, and students do not see it that way unless they are taught to.)

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  • $\begingroup$ I believe the OP was looking for finite-dimensional vector spaces, which rules out the space of functions. (It's an excellent example otherwise, though.) $\endgroup$ Commented Jun 17, 2021 at 0:19
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    $\begingroup$ @Steven Gubkin: Regarding #4 above, we can also consider $\mathbb R$ as a vector space over $\mathbb Q$, which is a seemingly never-ending source for counterexamples and interest in certain areas of real analysis and functional analysis and topology and set theory. (continued) $\endgroup$ Commented Jun 17, 2021 at 14:10
  • $\begingroup$ See also this 2 Oct 1999 sci.math post. $\endgroup$ Commented Jun 17, 2021 at 14:10
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    $\begingroup$ It's particularly noteworthy that $\mathcal{C}[a,b]$ is not a Hilbert space like $\mathcal{L}^2[a,b]$, even when equipped with the same inner product. As a consequence, you don't get the Riesz isomorphism between the space and the dual space of linear functionals on it (‘the space is not “the same” as its dual’), which is quite counter to $\mathbb{R}^n$ intuition. $\endgroup$ Commented Jun 18, 2021 at 8:55
  • $\begingroup$ Re 4, another thing commonly forgotten is that there is not "the" isomorphism between $\mathbb C$ and $\mathbb R^2$, but infinitely many. This goes so far that even when we agree on a field structure, "my" $i$ could be "your" $-i$. Also, in some contexts the base $\{\frac{1+i}{2}, \frac{1-i}{2}\}$ is more practical than everyone's favourite $\{1,i\}$ etc. $\endgroup$ Commented Mar 7 at 16:55
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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 is also an $\mathbb{R}[x]$-module, and that composition with polynomials from the right is a well-defined operation.

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  • $\begingroup$ I am sorry for waiting so long to comment, but this is indeed a very cute example. I wonder if it has any applications? $\endgroup$ Commented Jan 18 at 16:55
  • $\begingroup$ @StevenGubkin None that I know of, but perhaps it could have? $\endgroup$
    – Gaussler
    Commented Jan 18 at 17:11
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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 space.

This allows for an obvious extension, if you want to get into it: the set of solutions to an inhomogeneous system of linear ODEs is an affine space.

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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 $$ E \mathbin{\Delta} F = (E \cup F)\setminus (E \cap F) $$ as addition, and $$ 0E = \varnothing,\qquad 1E=E $$ as scalar multiple.


Any finite-dimensional vector space over $\mathbb F_2$ has cardinal $2^k$ where $k$ is the dimension. So we conclude any finite algebra of sets has cardinal $2^k$. In particular, without using this try to prove that there is no algebra of sets with $6$ elements.

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    $\begingroup$ More generally, this is true for any Abelian group where all objects square to the neutral element, such as your case of the power set with the symmetric difference as group operation. $\endgroup$
    – Gaussler
    Commented Jun 19, 2021 at 10:19
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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 these references because it's likely that people who google-land on this question in the future would be interested.

The first paper is freely available (see also this google search) and the second paper is behind a paywall.

Michael A. Carchidi, Generating exotic-looking vector spaces, College Mathematics Journal 29 #4 (September 1998), pp. 304−308.

Daniel S. Kalman and Prescott K. Turner, Algebraic structures with exotic structures, International Journal of Mathematical Education in Science and Technology 10 #2 (April−June 1979), pp. 173−174.

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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 $V$ is a vector space over $\mathbb{F}$, then so is $V^S$, again with pointwise addition and scalar multiplication.

  • One of my favourites (in case that we want to consider fields different from $\mathbb{R}$ or $\mathbb{C}$, too): The power set $2^S$ of a given set $S$ is a vector space over $\mathbb{F}_2 = \{0,1\}$, where addition is defined to be the symmetric difference, and $0 \cdot A := \emptyset$ and $1 \cdot A := A$ for each $A \subseteq S$.

    Actually, this space is isomorphic to $(\mathbb{F}_2)^S$.

  • A note on example 1. in the question:

    Again, if you consider general fields (and thus also fields of non-zero characteristic), polynomials even yield two classes of vectors spaces: polynomials in the "algebraic" sense (which are, strictly speaking, just lists of their coefficients) and polynomial functions.

    The mapping which maps each polynomial to its corresponding polynomial function (which is injective if and only if the field has characteristic zero) is a nice example of a linear map which is, maybe, not so easy to understand immediately in terms of matrices.

On a more general note, I'd say that one of the major motivations for making the abstraction from spaces such as $\mathbb{F}^n$ to general vector spaces is that - even when we restrict ourselves to the finite-dimensional case - general vector spaces quite are flexible in terms of constructing new spaces from given ones. So let me continue the list of examples by giving examples of "standard constructions":

  • As mentioned implicitly in previous answers, a vector subspace of a vector space is again a vector space in its own right. Thus, in particular, the kernel and the range of a linear map is a vector space (and hence, so is the set of solutions to a homogeneous linear equation).

  • If $V_1, \dots, V_n$ are vector spaces, then so is $V_1 \times \dots \times V_n$ (with componentswise addition and scalar multiplication.)

  • If $U$ is a vector subspace of a vector space $V$, then the quotient space $V/U$ is a vector space.

  • The dual space $V'$ of a vector space is a vector space (this is a special case of the space of all linear mappings between two given vector spaces - an example that was mentioned in a comment by James S. Cook).

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  • $\begingroup$ Is it possible your comment after your first example has an error in it? $\endgroup$
    – cheyne
    Commented Dec 26, 2023 at 13:25
  • $\begingroup$ @cheyne: I don't think so. Could you be more specific? $\endgroup$ Commented Dec 26, 2023 at 14:15
  • $\begingroup$ If I consider a vector space V and a set X and construct a vector space Set(X,V) from all functions from X to V, the basis is not isomorphic as a set to X, whether $V = \mathbb{F}$ for some field or not. Sure it has a basis for "reasons" but it didn't seem like that's what you were relying on. But if I take the free vector space, $\mathbb{F}^(X)$, which has a basis isomorphic to $X$ this feels more in line with the spirit of what you were implying. But maybe I misunderstood. $\endgroup$
    – cheyne
    Commented Dec 26, 2023 at 17:45
  • $\begingroup$ @cheyne: If $V$ is a vector space over $\mathbb{F}$ with basis $X$, then $V$ is isomorphic to the vector subspace of $\mathbb{F}^X$ that consists of those functions that vanish at all but finitely many points. That's why I wrote "isomorphic to a subspace". $\endgroup$ Commented Dec 26, 2023 at 18:00
  • $\begingroup$ I agree with that now that you wrote "vanish at all but finitely many points". Thank you! I think I was hoping you'd add that in after you wrote "mappings from S to F...". Because without that condition it's still a vector space but then the basis comment felt misleading. Thank you for helping me understand what you meant. $\endgroup$
    – cheyne
    Commented Dec 26, 2023 at 18:55
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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."

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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. These spaces actually have applications in graph theory. A purely graph-theoretic notion such as existence of a cycle has a corresponding linear algebraic notion in terms of linear endomorphisms, and sometimes useful in solving graph-theoretic problems. (Of course, these may be regarded as a special case of other examples.)

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Probably not considered mathematics but I think still useful for students:

  1. The space of possible velocities a particle in space can have
  2. The space of possible angular momenta a rigid body in space can have
  3. The space of forces that can act on a particle in space
  4. ...

These are abstract vector spaces in the sense that they are not supplied with a canonical basis and hence not canonically isomorphic to $\mathbb{R}^3$. These spaces are are also not canonically isomorphic to each other (unless someone believes that SI units are canonical.)

In general physical quantities deliver many more examples and I find it instructive to also mention one dimensional examples to students, like electrical charge.

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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-symmetric bilinear $k$-forms on $V$.

The advantage of these examples is functoriality; i. e., a linear operator on $V$ gives rise to linear operators on $V^\star,$ $\Lambda^k(V)$, etc., and this respects composition. From this observation, we readily get a coordinate-free definition of determinant, with the property that the determinant of a product is a product of determinants being automatic.

On a more elementary note, it is good to at some point introduce the students to the idea that the rows and columns of a matrix need not be indexed by integers (and in particular, that the rows and columns of an $n\times n$ matrix may be indexed by different sets.) This conveys some idea of usefulness of abstraction: while we can enumerate any finite set, this enumeration is arbitrary, and so is identification to $\mathbb{R}^n$. For an example, take an adjacency matrix of a graph, or a transition matrix of a Markov chain.

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Here's one of my favorite examples of an abstract vector space: $\mathbb{N}$ with the prime numbers as the basis and arithmetic multiplication as vector addition. For example, $5 = 2^0\cdot 3^0\cdot 5^1 \cdot 7^0 \cdot \cdots \longrightarrow [0,0,1,0,0,\ldots]$, and $24 = 2^3\cdot 3^1\cdot 5^0 \cdot 7^0\cdot \cdots \longrightarrow [3,1,0,0,0,\ldots]$.

The number $1$ becomes the zero vector, and two vectors being orthogonal means that the numbers are relatively prime. Scalar multiples of vectors are powers of the numbers.

We can expand the numbers to include all positive rational numbers and let the vector coefficients be all integers (positive and negative). For example, $\dfrac{14}{45} \longrightarrow [1, -2, -1, 1, 0, \ldots]$.

Note that technically this is a "module" rather than a true abstract vector space because we are taking our scalars to be elements of a ring ($\mathbb{Z}$) rather than elements of a field (like $\mathbb{Q}$ or $\mathbb{R}$).

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  • $\begingroup$ Quick follow-up: It looks like if you allow the scalars to be elements of $\mathbb{Q}$, then this would form a true vector space. The corresponding set of numbers would be bigger than $\mathbb{Q}_{>0}$, but smaller than $\mathbb{R}_{>0}$. My question is would the primes still form a basis? In other words, would the representation of a number in this set as a product of rational powers of primes be unique? If anyone has a thought, I'd love to know. $\endgroup$
    – Bill Evans
    Commented Dec 15, 2023 at 20:25
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Here is an example from Nelson's Radically Elementary Probability Theory (Zbl 0651.60001).

Suppose that $\Omega$ is a finite set with a strictly positive real function $p$ on $\Omega$ such that $\sum_{x \in \Omega} p(x) = 1$. So, $p$ is really a probability function on $\Omega$. A random variable $X$ is then just any function from $X$ to $\mathbb{R}$, and we define its expectation $\mathbf{E}[X]$ as $\sum_{x \in \Omega} p(x) f(x)$.

Now,

  1. the collection of all random variables, $\mathbf{R}^\Omega$, is a (finite-dimensional, real) vector space;
  2. $\mathbf{E}$ is a linear map into $\mathbf{R}$ (i.e., a linear functional on $\mathbf{R}^\Omega$). In particular, the collection of random variables with mean $0$ is a hyperplane $H$.
  3. $\langle X, Y \rangle := \mathbf{E}[XY]$ is an inner product on $\mathbf{R}^\Omega$;
  4. the orthogonal complement of $H$ with respect to this inner product is the space of constant random variables;
  5. the typical way of shifting an arbitrary random variable $X$ to one with mean $0$ by considering $X - \mathbf{E}[X]$ is the same thing as taking the projection of $X$ onto the subspace $H$;
  6. similarly, computing the mean is the same as finding the length of the projection onto the orthogonal complement of $H$, i.e., onto the 1D space of constant random variables;
  7. for a mean $0$ random variable $X$, the standard deviation is just its length;
  8. for mean $0$ random variables $X$ and $Y$, the correlation coefficient is the cosine of the angle between them.

Lastly, the radically elementary idea is that this nice picture can be very useful even when $\Omega$ is not finite, via nonstandard methods.

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    $\begingroup$ This is very cool, and quite relevant to the kind of work I am doing now! $\endgroup$ Commented Mar 15 at 17:41
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Something else that I haven't seen explicitly:

Let $k$ be a field. A $k$-algebra $A$ with ring homomorphism $\phi:k\to A$ is a $k$-vector space with scalar multiplication: $$\lambda \cdot a=\phi(\lambda)\cdot a$$ (where $\lambda\in k, a\in A$).

A quite cool example:

Let $A$ be the following ring (known as the Weyl Algebra): $$A=\mathbb{C}\langle x,y\rangle/\langle yx-xy-1\rangle$$ If you haven't seen this before, $\mathbb{C}\langle x,y\rangle$ is the free algebra with basis in letters $x,y$. This is similiar-ish to $\mathbb{C}[x,y]$ except that the variables don't commute: $$xy\ne yx$$ You can have something like: $$(3i)xy+\sqrt{2}y^2 x- yxy+\cdots$$ Anyway, by quotienting by the free algebra generated by $yx-xy-1$ we have the relation: $$yx=xy+1$$

There's some fun stuff in there.

Algebras like these are closely related to differential operators, and differential equations, and some have shed light on certain partial differential equations. So what be seen as weird, has really important application.

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If you have a rigid body, you may apply a force to it, and if you apply two forces at the same point, they are equivalent to their sum applied at that point, but this is not the case for forces applied at different points. So, mechanics textbooks implicitly consider the (infinitely-dimensional) vector space $V$ which is a direct sum of uncountably many copies $\mathbb{R}^3$, indexed by points in the body.

Now, this vector space has a (still infinite dimensional) subspace $V_0$ of forces that do not cause the body to move - for example, when two persons both pull a table in the opposite directions. The laws of statics describe the basis of $V_0$, while the theory of torque deals with constructing an explicit isomorphism $V/V_0\mapsto \mathbb{R}^6$.

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