Skip to main content

Why A^{T}A is invertible? (4) Linear Algebra

Appendix A: null space and column space
We use the null space for the proof, therefore, I will explain the null space a bit. If you know about the null space, of course you can skip this entry.

The null space of a matrix A is a set of non-zero vector x that satisfies Ax = 0.
Let me show you an example square matrix A that has null space.
When x \neq 0 , following x is a solution.
Therefore, this x is a null space of A. When an a is an scalar, ax, a \neq 0 are also the solutions. It means these are also null space. In this example, the matrix is singular (and square). If the matrix is not singular, the solution must be 0 only. Because, if a square matrix is not singular, there is the inverse,
Therefore, x = 0. In this case, we say there is no null space.

Let me show you another example, but this time a rectangle matrix A that has null space.
The solution is the same as the last example.

By the way, these are all about definition of null space. I could finish the story about null space at this point, however, I would like to think about the meaning of null space a bit more. I would like to talk about the existence of the inverse matrix, so let's talk about square matrices. Because a rectangle matrix can be interpreted as a matrix that has too many equations or too less equations, there is no inverse for such matrices.

null space is related with a matrix is singular or not, that means the matrix has independent columns or not. Because, if we write down an A with column vector, and multiply with x,

Therefore, Ax is a linear combination of the column vectors a_i with coefficients x_i. If this result is 0, x is a null space. This is one aspect of null space.

When we look at the columns of Ax = b, b is the linear combination of column vector of A. A space consists of column vectors is called column space. If we have n independent columns in n-dimensional space, we could express any point in the n-dimensional space. This means we have a solution for any b and we have the inverse. If A has the inverse, the linear combination of the column vector becomes 0 iff x= 0. This also means there is no null space (= or only 0 exists).

Now we see the null space, column space, and the existence of the inverse. Do you think you have now a bit clearer view about the relationship among them?

Acknowledgements

Thanks to Marc D. who listened my explanation and gave me comments.

Comments

Popular posts from this blog

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides:  A bit intuitive (for me) explanation of area of parallelogram  (to my site, external link) .