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?
In: 1
You can consider the columns of a matrix as a **set** of (column) vectors. The space they span is the column space.
For example the matrix
1 2
3 4
5 6
has two column vectors, (1 3 5) and (2 4 6), which span a two-dimensional subspace of **R**^3
The null space is just the set of vectors going into 0 when you apply the matrix on them. If A is the matrix and x the vector, the null space consists of all vectors solution of the equation
A.x = 0
It’s easy to show that they form a subspace (of the space where you take your x-es).
Do you know how to multiply a matrix by a vector?
For an m x n matrix, its column space is *all* the linear combinations of the columns. What that means in terms of matrix-vector multiplication is that the column space is *all* the vectors A**v** as **v** runs over **R**^(n). In other words, if you use the matrix A to define a function from **R**^(n) to **R**^(m) that sends each **v** to A**v**, then the column space is the range of that function.
The null space is all the **v** where A**v** is the zero vector. It tells you how much A “collapses” **R**^(n) when you apply it to vectors there: A**u** = A**w** exactly when A(**u** – **w**) is the zero vector, meaning **u** – **w** is in the null space of A.
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.
Latest Answers