The Dimension of a Vector Space

The dimension of a vector space is simply the number of vectors in a basis for it, which is the number of independent directions you can move in. It matches the everyday word: a line is 1-dimensional, a plane is 2-dimensional and the room around you is 3-dimensional. What's new is that it works for any vector space, so it makes sense to say $\mathbb P_2$ is 3-dimensional, because it takes three numbers to pin down $a + bt + ct^2$. That's useful because once you know the dimension, just counting vectors tells you a lot. All we're really saying in the formula is that $\dim V$ (the dimension of $V$) is a count of basis vectors, and the word "any" (every basis gives the same count, so it doesn't matter which one you picked) is what makes that count trustworthy. It's a count, nothing scarier; bear with it.

Why it's well defined. In a space spanned by $n$ vectors, any set of more than $n$ vectors is dependent. Now take two bases. Each one spans, so neither can have more vectors than the other without being dependent. So two bases can't have different sizes.

Consequences in an $n$-dimensional space:

• Any $n$ independent vectors automatically form a basis.
• Any $n$ vectors that span automatically form a basis.
• More than $n$ vectors are always dependent; fewer than $n$ never span.

It makes sense that you get one check for free. If $n$ independent vectors failed to span, you could add a vector outside their span and have $n + 1$ independent vectors, which the space doesn't allow.

Subspaces of $\mathbb R^3$ come in dimensions 0 (the origin), 1 (lines through the origin), 2 (planes through the origin) and 3 (all of $\mathbb R^3$).

A common mistake. It's tempting to think the plane $x + y + z = 0$ has dimension 3, because its points have three coordinates. It isn't: dimension counts the directions you can move within the space, not the size of the vectors. Once you choose $y$ and $z$, the equation decides $x$, so only two numbers are free. There's an animation further down that shows two weights locating every point on that plane.

"Degrees of freedom" in physics and engineering is dimension: a rigid body in space has 6 (3 translations, 3 rotations), and a robot arm needs at least 6 joints to reach any pose.