Can anyone please explain the the difference between a vector and a matrix?

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?

Solution 1:

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).

Solution 2:

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.