What is ${\displaystyle \mathbb {R} ^{n}}$?

Question

When I first started learning linear algebra we are told that ${\displaystyle \mathbb {R} ^{n}}$ is the set of n-tuples. When studying calculus of $n$ variables we would often use a function of a Matrix $M$ and a column vector $y$ with $n$ rows and we would say $y ∈{\displaystyle \mathbb {R} ^{n}}$, again $y$ being a column vector it is in fact a matrix, although it could in this case be representing a list of numbers, but in either case, neither of these are 'tuples' so how are they elements of ${\displaystyle \mathbb {R} ^{n}}$?

I understand the idea of representing a tuple as a column vector, in which case are column vectors elements of ${\displaystyle \mathbb {R} ^{n}}$? Is this the case, if a column vector as a 'list of numbers' is an element of ${\displaystyle \mathbb {R} ^{n}}$, is a row vector?

Answer 1

Elements of $\def\Rn{\Bbb R^n}\Rn$ are $n$-tuples of real numbers. The does not really answer the question unless one knows what an $n$-tuple really is, but as long as we have a standard format for writing down mathematical values, one usually does not ponder too long about what the values really are; things like of numbers, matrices, polynomials. The important things are being able to decide what we are allowed to write down, and under what condition two such expressions denote the same value; for instance for a rational number we are allowed to write a fraction of two integers with the bottom one being nonzero, and two such fractions denote the same rational number if cross-multiplication gives two equal integers. By the way, such an approach does not really work for real numbers, since we cannot actually write most of them down, and for the kind of expression that we do have to denote them, the rules for deciding equality can be quite subtle or even undecidable; however this point is usually sidestepped, and we pretend to be able to write down real numbers (with examples invariably involving values that we can indeed write down). So an $n$-tuple of real numbers is an object that assigns to each of $n$ component positions (usually numbered $1,2,\ldots,n$ for convenience) a real number; it is usually denoted by writing those $n$ values separated by commas and enclosed in parentheses, and two such expressions denote the same tuples if and only if for each component positions the assigned values are equal as real numbers. If this sounds overly pedantic then maybe it is, but if you ask what $\Rn$ really is, then you are bound to get into this kind of consideration.

Now there are many similar objects that correspond in an obvious way to $n$-tuples of real numbers. We could allow square brackets to be used instead of parentheses, and/or spaces instead of commas, for instance. We could stack the positions vertically instead of horizontally, as long as we are clear about the (total) order from first to last. Under this form one could say that $n$-component column vectors are (disguised) $n$-tuples in $\Rn$, and since they look the same we can easily forget the distinction between these and $n\times 1$ matrices, which is particularly appealing because the multiplication of a matrix with an element of $\Rn$ and that of the same matrix with a $n\times 1$ matrix proceeds in precisely the same manner.

Vectors in any real vector space of dimension$~n$ that are not themselves $n$-tuples of real numbers do not correspond to elements of$~\Rn$. This applies in particular to elements of euclidean vector spaces, for which a geometric description is often used. However, if a basis of such a vector space is fixed (which is an ordered sequence of vectors), then the coordinates of any vector with respect to this basis form an element of $~\Rn$. Some vector spaces (other than $\Rn$) come with "everyone's favourite" basis, such as the basis $[1,X,X^2,\ldots,X^{n-1}]$ for the space of real polynomials of degree less than $n$. In such cases it is tempting to use the corresponding coordinate map to "identify" vectors with their coordinate tuples; however it is usually best not to do that, and maintain the distinction between vectors and their coordinates. (In $\Rn$ we have no such luxury, since its elements are $n$-tuples of real numbers.)

Answer 2

Of course a euclidean vector is itself not a tuple, but a geometric object in its own right.

I think that knowing about affine spaces (or euclidean spaces, affine spaces with an inner product on the translation space) will solve your problems. Indeed, they are used in modern and rigorous treatments of euclidean geometry, see this book for example. The choice of an origin allows us to identify each point with its position vector and the choice of a basis allows us to identify each vector with a column vector, i.e. an element of $\mathbb R^n$.

Answer 3

Let $A, B$ be two sets.

Then $B^A$ denotes the collection of all functions $f:A\to B$ .

Let $A$ is a finte set with $n$ elements and identify $A$ as $\{1,2,...,n\}$ and $B=\Bbb{R}$.

Then $\Bbb{R}^n$ is the collection of all functions $f:\{1,2,\ldots,n\}\to\Bbb{R}$

$$\Bbb{R}^n=\{(f(1),f(2),\ldots,f(n)) : f:\{1,2,\dots ,n\}\to\Bbb{R}\}$$

Let $f(k) =a_k\space ,\forall k\in\{1,2,\dots,n\}$

Then $$\Bbb{R}^n=\{(a_1,a_2,\dots,a_n):a_k\in \Bbb{R},1\le k\le n\}$$

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Every elements of $\Bbb{R}^n$ is an $n$-tuple.

$V=(\Bbb{R}^n, +\cdot) $ forms a vector space over $\Bbb{R}$ .An element of $V$ has $n$ components which are real numbers. If we list in a row, we get a $n$ tuple of reals (row vector or $1×n$ matrix ) and of list in a column we get a column vector or a $n×1$ matrix both are members of euclidean space with two different representation.

$v=\begin{pmatrix}v_1&v_2&\ldots&v_n\end{pmatrix}$

$v=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$

Answer 4

The notation / meaning is indeed somehow ambiguous. There are many conventions to which people got used. I would say that $\mathbb{R}^n$ usually denotes the pair

• Set of all n-tuples of real numbers, and
• Some additional structure on this set, such as vector, affine, metric, euclidean, topological or other space, depending on the context.
  • Unless explicitly mentioned, all of these structures are "standard", for instance metric distance would be $\sqrt{\sum (x_i - y_i)^2}$.
  • If explicitly mentioned, people also consider non-standard structures on this set, such as discrete or Zariski topology, or a Symplectic vector space

To get back to your original question, there is not really a big difference between a "column vector" and "row vector". If you consider this to be a tuple of numbers, then it is only a matter of how you write these numbers.

(Of course, if you introduce some formalism on matrix multiplication, you need to choose the row / column formalism correctly so that things make sense. Very often $v\in \mathbb{R}^n$ would be a "column vector" and $v^T$ a "row vector". In this convention, we somehow identify vectors with matrices $n\times 1$.)

Answer 5

$R^n$ is indeed the set of all n-tuples. The reason some people refer to column vectors as elements of $R^n$ are as follows.

In Linear Algebra, a matrix is used to represent a certain function called a linear map or linear transformation from a vector space $V$ to another vector space $W$. Linear maps come first and matrices come second. Most people are introduced to linear algebra through matrices and that is how people like doing mathematics with linear algebra.

Consider the function $T : R^2 \to R^2$ defined by $T(x, y)=(2x, 2y)$. The vector $(1, 2)\in R^2$. Thus, we can evaluate this vector for the function $T$. $T(1, 2)=(2, 4)$.

The above function is a linear map. Given that we can evaluate the vector using this function, we would also like to be able to evaluate the vector with the matrix that represents this linear map. But it makes no sense to multiply a matrix by a tuple. Thus, like we defined the matrix of a linear map, we also define the matrix of a vector.

The matrix of a n-tuple in $R^n$ is defined to be the $n\times1$ matrix whose rows consist of the scalars needed to write the n-tuple as a linear combination of the basis of $R^n$.

For example: The vector $(1, 2)\in R^2$. This vector can be written as a linear combination of the basis $(1, 0), (0, 1)$ as $1(1, 0)+2(0, 1)$. The matrix of this vector is the $2\times1$ matrix that consists of the scalars needed to write it as a linear combination of the basis of $R^2$. Let $v$ denote the matrix of the vector $(1, 2)$. Then, $v=\begin{bmatrix}1 \\ 2\end{bmatrix}$

Now we come to the main part. Let $M(T)$ denote the matrix of the linear map $T$. Let $M(v)$ denote the matrix of a vector $v\in R^n$. Then, a result in linear algebra tells us the following.

Theorem: $M(T(v))=M(T)M(v)$. Namely, evaluating a vector using a linear map is equivalent to multiplying the matrix of that vector with the matrix of the linear map. The result of this is the matrix of $T(v)$.

Example: We saw that $T(1, 2)=(2, 4)$. The matrix of $T=\begin{bmatrix}2 & 0\\0 & 2\end{bmatrix}$. The matrix of the vector $(1, 2)=\begin{bmatrix}1 \\2\end{bmatrix}$.

Then, we see that $\begin{bmatrix}2 & 0\\0 & 2\end{bmatrix} \begin{bmatrix}1\\2\end{bmatrix}= \begin{bmatrix}2\\4\end{bmatrix}$.

The resulting vector can be written as a tuple as well. Namley, just use the definition of the matrix of a vector backwards.

This is the reason a lot of people refer to column vectors as elements of $R^n$. As a matter of fact, you can make a vector space of the column vectors of $R^n$. This is using the operations of addition and scalar multiplication of matrices. Then, this new vector space and $R^n$ are isomorphic. Namely, the map that takes every vector $v\in R^n$ to its matrix $M(v)$ in this new vector space.

I am sure you have plenty of more questions. Such as, how do we define the matrix of a linear map, what happens if I change the basis in the definition of the matrix of a vector, does the definition of the matrix of a vector hold on other vector spaces and a few more. These will take some time to explain and the point of the answer was to tell you the reason why people refer to matrices of vector as elements of $R^n$. To answer all the other question, I would recommend reading Linear Algebra Done Right by Sheldon Axler. He discusses all of these things. As a matter of fact, that is where I learnt it all of this first.

Hope this answer helps!

Answer 6

If we fix a basis, e.g. the usual orthonormal basis, there is $1-1$ relationship between $\mathbb{R}^{n}$ as a vector space and $\mathbb{R}^{n}$ as a set of n-tuples. Which implies they are identical mathematical objects. But this is true ONLY when we fix a basis in the first place. Otherwise an n-tuple is meaningless in a vector space, whereas it makes sense in the $\mathbb{R}^{n}$ of Analysis! Because it means that is just an element of $\mathbb{R}\times\mathbb{R}.....\times\mathbb{R}$.!

Answer 7

Here are some small hints and if you like a word of warning. If you are asking: - what is $\mathbb{R}^n$ - then you are asking about $\mathbb{R}^n$ as a set.

• The answer: $\mathbb{R}^n$ is the set of $n$-tuples of real numbers - is fine, since the elements are $n$-tuples. They are plain ordered lists of $n$ real numbers and nothing else.

The last three words are some kind of warning. It is not appropriate to call an element vector as per se there is no vector space structure associated with $\mathbb{R}^n$. It is just a plain set. The elements do not have any additional structural properties, they are like dust without additional shape, not carrying any additional information.

• Of course, you are free to add additional structures to the set $\mathbb{R}^n$ making it a group or a vector space or a topological space or whatever you like as long as it is mathematically admissible. But only after introducing an additional structure, the elements might be not only $n$-tuples, but different objects, points, vectors, etc.

• The essence of an $n$-tuple is the number $n$ of components and the ordering of its components. We might organize these elements as row or as column or as something else, - this is not essential and just a matter of convenience.