I’m just getting started with linear algebra and I’m not sure what column spaces or null spaces actually signify. What is the geometrical interpretation of both?
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Ok. all these answers are telling you how to calculate it, but you wanted a geometric interpretation, so bear with me.
Matrices are as incredibly useful as they are because they give us a concise way of representing *linear transformations*. You can think of it as a function from one vector space to another.
Lets make it concrete and say that a matrix **A** is a way of taking a vector in vector space **V** and returning a vector in vector space **W**. The columns of **A** are themselves vectors in **W**, and the *column space* of **A** is just the space of all vectors in **W** you can get to by applying **A** to a vector in **V**.
For example, say **A** is a projection from R^3 to R^2. Then the column space is just R^2 itself, since you can get any 2D point (x, y) by projecting a 3D vector (x, y, z) to it. On the other hand if **A** takes you from R^2 to R^3 by rotating the plane 45 degrees about the x-axis, then your column space is that rotated plane, since every point you can get with **A** lies on this rotated plane. Sure, the plane is embedded in a larger space R^3, but the rest of the vectors in R^3 are “inaccessible” by **A**.
Every linear transformation maps 0 to 0. The null space of A is the set of *all other* vectors in **V** that get mapped to 0. Say **A** is a projection from R^3 to R^2, like in our first example. then the null space is precisely the vectors on the z axis in **W**. You can think of the dimension of the null space as a measure of how many axes of information is “lost” when taking the linear transformation.
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