# What is the Dimension of the Null Space for Matrix A?

• snoggerT
In summary, the dimension of the null space of the given matrix A is 1. This can be found by solving the linear system u+3v=0 and -2u-6v=0, which can be expressed in terms of one parameter. Alternatively, the dimension can be found by subtracting the rank of A from the number of variables.
snoggerT
Find the Dimension of the null space of the given matrix A:

| 1 3|
|-2 -6|

## The Attempt at a Solution

I honestly don't know how to work this at all. I think I'm confused as to what Null Space actually is, so that's making this a difficult problem to understand. please help.

A vector x is in the null space if Ax=0. If you write x as the column vector (u,v) then you want to solve the linear system u+3v=0 and -2u-6v=0. You must have done something like that before, right? How do you do it?

will using this work...

# of variables - Rank(A)= dim(null space A) ?

using that formula gave me the correct answer.

Sure. If you found the rank of A=1 then the dimension of the null space is 2-1. You could also have solved the linear system to find the answer can be expressed in terms of one parameter. That also means the dimension of the null space is one.

the dimension is just how many vectors you get out of the kernel.

if there are 2 variables and the rank is 1 (one leading 1) , there must be 1 free variable, which means the dim will be 1 and there will only be 1 vector for the kernel.

that makes a lot more sense to me now. thanks.

I really dislike using formulas like that when it can be done straight from the definitions:

v is in the null space of A if and only if

$$\left(\begin{array}{cc}1 & 3 \\-2 & -6 \end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)= \left(\begin{array}{c}0 \\ 0\end{array}\right)$$
which is the same as saying x+ 3y= 0 and -2x- 6y= 0. What (x, y) satisfy both of those equations?

## 1. What is the dimension of the null space?

The dimension of the null space is the number of linearly independent vectors that span the null space of a matrix. It can also be referred to as the nullity of a matrix.

## 2. How is the dimension of the null space related to the rank of a matrix?

The dimension of the null space and the rank of a matrix are related by the rank-nullity theorem, which states that the sum of the rank and nullity of a matrix is equal to the number of columns in the matrix.

## 3. How can the dimension of the null space be calculated?

The dimension of the null space can be calculated by finding the rank of the matrix and subtracting it from the number of columns in the matrix. Alternatively, it can also be calculated by finding the number of free variables in the reduced row-echelon form of the matrix.

## 4. What does a larger dimension of the null space indicate?

A larger dimension of the null space indicates that there are more linearly independent solutions to the homogeneous system of equations represented by the matrix. It also means that the matrix has fewer linearly independent columns.

## 5. How does the dimension of the null space affect the invertibility of a matrix?

The dimension of the null space is related to the invertibility of a matrix. If the dimension of the null space is zero, the matrix is invertible. However, if the dimension of the null space is greater than zero, the matrix is not invertible and is considered singular.

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