# Why null space and column space?

I am not asking this question for WHAT is null space or WHAT is column space. I have finished learning about the definitions of these two concepts for a while. However, to install these concepts in my mind forever, I really want to know what the purposes are for null space and column space of a vector.

Thanks!

• Null space. They are the solutions to the equation $Ax=0$ where A and x are matrices. It's like asking why is $x= \frac{-b +-\sqrt{b^2-4ac}}{2a}$ solutions important. Commented Oct 23, 2014 at 8:29
• math.stackexchange.com/questions/21131/… This will be helpful. Meaning of Null space is asked here. Commented Oct 23, 2014 at 8:42
• I think you mean "null space and column space of a matrix," Justin. Commented Oct 23, 2014 at 9:19

Perhaps an example will clarify things.

Let's suppose that the matrix A represents a physical system. As an example, let's assume our system is a rocket, and A is a matrix representing the directions we can go based on our thrusters. So what do the null space and the column space represent?

Well let's suppose we have a direction that we're interested in. Is it in our column space? If so, then we can move in that direction. The column space is the set of directions that we can achieve based on our thrusters. Let's suppose that we have three thrusters equally spaced around our rocket. If they're all perfectly functional then we can move in any direction. In this case our column space is the entire range. But what happens when a thruster breaks? Now we've only got two thrusters. Our linear system will have changed (the matrix A will be different), and our column space will be reduced.

What's the null space? The null space are the set of thruster intructions that completely waste fuel. They're the set of instructions where our thrusters will thrust, but the direction will not be changed at all.

Another example: Perhaps A can represent a rate of return on investments. The range are all the rates of return that are achievable. The null space are all the investments that can be made that wouldn't change the rate of return at all.

Another example: room illumination. The range of A represents the area of the room that can be illuminated. The null space of A represents the power we can apply to lamps that don't change the illumination in the room at all.

Good luck!

• best answer ever :D Commented Oct 24, 2014 at 1:37
• @JustinChan Thanks man! If you'd like to know more about applications of Linear Algebra like the ones I've described, reviewing the online lectures of Linear Dynamical Systems by Stephen Boyd of Stanford may be of interest after your Linear Algebra class. Commented Oct 24, 2014 at 1:44
• by the way, I was looking at some questions that are quite similar to mine, and I used your answer as a quote! Hope you don't mind :D Commented Oct 24, 2014 at 1:58
• @JustinChan Of course not! It's very flattering. I'm happy to help. Commented Oct 24, 2014 at 3:03
• According to your explanation, there's no way Range(A) could possibly equal to Null(A), right? I also tried thinking if it's possible for Range(A) = Null(transpose(A)) I think the answers to both questions are no. Commented Oct 24, 2014 at 21:11

Basically, you want to know how the null and column spaces are applied. One of the applications is in robotics field. Let $$x\in\mathbb{R}^n,q\in\mathbb{R}^m$$ represent the pose (i.e. position and orientation) of a manipulator's end-effector and its joints' positions. The kinematic equation is given as follows $$x = f(q). \tag{1}$$
where $$f : \mathbb{R}^m \rightarrow \mathbb{R}^n$$. The differential kinematic equation is obtained by taking the time dervative of Eq.(1); therefore, we get $$\dot{x} = J(q) \dot{q} \tag{2}$$ where $$J(q) = \dfrac{\partial f(q)}{\partial q}$$ the Jacobian matrix (i.e. very important matrix in robotics). When a robot has more joints than is required for a particular task, it is classified as redundant. Specifically, we are interested in moving the arm without impacting the end-effector motion of the robot (i.e. $$\dot{x} = 0, \dot{q} \neq0)$$. Such configurations exist in the null space of the Jacobian matrix (i.e. $$\dot{q}\in N(J))$$. The configurations of the Jacobian matrix must belong to the column space if we wish to move the end-effector.