 # How to find the null space and the nullity of a matrix: Example

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Is an example of how to find the null space of a matrix by definition. Definition. The null space of a contains vectors.
X. That are solutions to this homogeneous system. A times x equal to the zero vector.
So in this example. Were solving the homogeneous system. With the coefficient matrix.
1. 2. 1.
1. 1. 2.
3. 5. 4.
1. 4. Negative.
1. The unknown. Vector.
X. Being. X.
Sub. 1. X.
Sub. 2. X.
Sub. 3 x. Sub.
4. And the right hand side vector is the zero vector 0 0. 0.
To solve this system we form the augmented matrix a railroad is lit to its ashland form. So this augmented matrix is real equivalent to its ashland form. Which is 1 0 0.
0. 1 0 2 1 0 3 negative 2 0 0 0 0.

From this augmented matrix we can tell that the general solution to this homogeneous. System. Satisfies.
X. Sub. 1.
Plus. 2. X.
Sub. 3. Plus.
3. X. Sub.
4. Equal. To.
0. And. X.
Sub. 2. Plus.
X. Sub. 3.
Minus. 2. X.
Sub. 4. Equal.
To. 0. So.
The general. Solution. Is x.
Sub. 1. Equal.
To negative. 2 x.

Sub. 3 minus three x sub. 4.
Where x sub. 1. Is a dependent variable x sub 3 x sub 4 are independent variables and the other dependent variable x sub.
2. Can be expressed from this equation as negative x sub 3. Plus.
2 x. Sub. 4.
So. The null space of a is a set of vectors that are four by one whose components. Except one x.
Up to x sub 3 x. Sub. 4.
Satisfy these two relationships. Only it contains all those vectors x. That a federal.
One whose components can be written as x. Sub. 1.
Is equal to negative 2 x. Sub. 3.
Minus. Three. X.
Sub. 4. X2.
Is. Negative. X.
Sub. 3. Plus.
2. X. Sub.
4. And x sub.

3. Is free to take any values and so is exit. 4.
So the last phase of a contains federline vectors x. Whose first two components can be expressed in terms of its third and fourth components. In such a way and to write a space in a nicer way.
So that its structure is more clear. We want to write this vector as a linear combination of a set of vectors. So from this expression.
This vector is equal to negative 2 x. Sub. 3.
Minus. Three. X.
Sub. 4. Negative.
X. Sub. 3.
Plus. 2. X.
Sub. 4. X.
Sub. 3. Zero.
Times. X. Sub.
4. Zero. Times.
X. Sub. 3.
X. Sub. 4.
Which means. That is equal to the sum of two vectors.

Negative. Two x sub. 3.
Negative. X. Sub.
3. Except. Three zero.
Times. X. Sub.
3. Plus. Negative.
3. X. Sub.
4. 2. X.
Sub. 4. Zero.
Times. X. Sub.
4. And x. Sub.
4. Those that were splitting first expression into two halves from where we can software out x sub. 3.
And s sub. 4 from each vector so the first vector is accept 3 times negative. 2 negative 1 1 0 x sub 4 times negative 3 0 1.
So then the most face of matrix a can be written as this set that contains all vectors x. That can be expressed as a linear combination of the two vectors negative 2 negative 1 1 0 and negative 3 2 0 1 and an electro satisfied this condition from the mouse base of a so you can see that so you can see that in those muscle a is the span of these two vectors negative. 2 negative 1 1 0.
And negative 3 2 0 1. In fact these two vectors form a basis for the mass goes away and because this basis. Contains two vectors.
The manatee of a which is the dimension of the null space of a is equal to 2. .