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I just took Calculus 3 last semester at my University and got comfortable with the idea of vectors, vector-valued functions, and basic vector operations like the dot and cross products.

This semester, I'm taking Differential Equations and we seem to be throwing around the terms "vector" and "matrix" as if they're interchangeable, especially now that we're studying systems of first-order differential equations. Additionally, there's mention of "vector spaces" which haven't been clearly explained to me.

The last time I dealt with matrices was in Algebra II in the 9th grade about 4 years ago, so there's quite a disconnect here. I feel like if I had taken Linear Algebra, this course would have been easier since my professor keeps saying "if you've taken Linear Algebra, then this should be familiar to you" which isn't exactly helpful.

Can anyone help me bridge these gaps in my understanding?

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    $\begingroup$ vector is a skinny matrix. $\endgroup$
    – Memming
    Nov 19, 2015 at 13:25
  • $\begingroup$ Vector spaces are the main objects of study in linear algebra, so you'll need to study that to understand the relationship between matrices and vectors. My suggestion: find some short intro to linear algebra and start working through it. $\endgroup$
    – user137731
    Nov 19, 2015 at 13:29
  • $\begingroup$ Can you give a more specific example of how the terms "vector" and "matrix" are being used as if they're interchangeable? $\endgroup$
    – littleO
    Nov 14, 2017 at 3:58

4 Answers 4

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Very roughly speaking ...

A matrix is a 2-dimensional rectangular array of numbers. If the array has $m$ rows and $n$ columns, we say that we have a matrix of size $m \times n$.

A vector can be regarded as a special type of matrix. A row vector is a matrix of size $1 \times n$, and a column vector is a matrix of size $m \times 1$.

You probably know how to multiply matrices. Since vectors are just special types of matrices, you know how to multiply a matrix times a vector. Multiplying by a matrix is often used as a way to somehow "transform" a vector (to rotate it or mirror it or scale it, for example).

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  • $\begingroup$ This makes it sound like matrices and vectors are both just arrays of numbers. This completely discards the more geometric interpretation. A vector is an element of a vector space. Only when you choose a base for this vector space, is the vector mapped to an array of numbers - and choosing two different bases will result in two different arrays of numbers for the same vector. $\endgroup$
    – Stef
    Dec 12, 2022 at 9:39
  • $\begingroup$ Yes, it does. In many scenarios (including the OP’s, I suspect) you just choose a fixed basis and stick with it. So then a vector is just a list of numbers. He’s doing differential equations. $\endgroup$
    – bubba
    Dec 13, 2022 at 11:14
  • $\begingroup$ And yet these numbers actually represent something. They're not just random numbers. And solving equations very often require changing bases, for instance picking the basis of eigenvectors of a matrix. These change of bases operations can be extremely obscure if you just think of them as abstract manipulations of lists of numbers. Or they can be extremely intuitive if you think of them from a geometrical point of view, and think of vectors as vectors, i.e., elements of a vector space, rather than as lists of numbers. $\endgroup$
    – Stef
    Dec 13, 2022 at 11:20
  • $\begingroup$ I insist, because when I read your comment, it sounds a lot like "The OP is just doing differential equations, so they don't need to understand what they're doing.". I know that it's not what you're saying, but it's really what it sounds like. Saying that vectors are "just lists of numbers" is not taking the easy route, it's taking the "I just want to manipulate numbers without understanding what I'm doing" route, which actually makes everything much harder. $\endgroup$
    – Stef
    Dec 13, 2022 at 11:23
  • $\begingroup$ @Stef. If you don’t like my answer, I suggest that you downvote it (if you haven’t already), and consider writing a better one of your own. $\endgroup$
    – bubba
    Dec 16, 2022 at 12:23
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There are canonical bijections:

$$ \mathbb{R}^n \cong \mathbb{R}^{1\times n} \cong \mathbb{R}^{n\times 1}$$

You can think of an $n$-component vector as a $1\times n$-matrix or as a $n\times 1$-matrix. People then like to pretend all the above sets are actually equal.

The advantage of this identification is that matrix multiplication can then be done with vectors or vectors and matrices without an extra definition.

On the other hand: I pretended "vector" means "tuple". Obviously vector spaces can contain all sorts of things. In particular, there is always an $\mathbb{R}$-vector space of $\mathbb{R}^{m\times n}$ matrices. But usually people don't call these matrices vectors, as far as I know.

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    $\begingroup$ Since the OP doesn't understand matrices and vectors, he (or she) probably won't find "canonical bijections" very helpful. $\endgroup$
    – bubba
    Nov 28, 2015 at 9:35
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    $\begingroup$ @bubba There is an (I believe) easy explanation of the fact underneath said statement. If the OP finds your answer more helpful, that's totally fine. But I will leave mine here anyway. Sometimes, I'd rather make an answer that benefits alot of people, not only the OP. $\endgroup$
    – Stefan Perko
    Nov 28, 2015 at 20:57
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Vector is 1 dimensional, so it is either only in "column" or "row" form. Matrix is 2 dimensional, it has both columns and rows and called m*n matrix(m columns and n rows). There is also "list" that includes both vectors and matrices and also other data formats(character formats - you will need this terminology in R)

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  • $\begingroup$ This question already has an accepted answer from $3$ years ago, please instead contribute to more recent questions. $\endgroup$
    – user635953
    May 15, 2019 at 14:27
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Fundamentally, vectors and matrices are different things.

A vector, e.g., $\mathbf{v} \in \mathbb{R}^n$, is a numerical entity in an $n$-dimensional space. A matrix, e.g., $\mathbf{A} \in \mathbb{R}^{m \times n}$, is a linear transformation from a $n$-dimensional to a $m$-dimensional space. In other words, if $T\left\{\cdot\right\}$ is a linear transformation, then there exists a matrix $\mathbf{A}$ such that $T\left\{\mathbf{v}\right\} = \mathbf{Av} = \mathbf{w} \in \mathbb{R}^m$.

  • So why so many people states that "vector is just an one-column matrix"?

Some points:

  1. Many people, actually, don't even have the insight that $\mathbf{Av}$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$.
  2. Denoting a vector as a column is a mere convention, indeed. Consequently, they are prone the think a vector as a particular case of a matrix when there is only one column.
  3. My definition holds only for linear algebra analysis. For multilinear algebra, a matrix (or a tensor) is a numerical entity as well as a vector.

So, in nutshell, vectors are numbers, and matrices are how these numbers are (linearly) transformed. A good reference the book of the renowned Professor Gilber Strang. In section 7.2, he discusses that a matrix is actually a linear transformation. I seize this difference only when I could see (literally) how matrix transforms spaces, on the Youtube videos of 3blue1brown, that is utterly beautiful.

I am not saying that seeing a vector merely as a one-column matrix is wrong though. But, IMHO, it is a shallow interpretation.

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  • $\begingroup$ A row vector describes a linear map $\Bbb R^n\to\Bbb R$, a column vector is, via scalar multiplication, a map $\Bbb R\to \Bbb R^n$, so your distinction has a difference of zero. $\endgroup$
    – Lutz Lehmann
    Dec 12, 2022 at 6:02

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