# Linear Algebra : Range and Null Space of a Matrix

## Example Questions

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### Example Question #1 : Range And Null Space Of A Matrix

Calculate the Null Space of the following Matrix.

There is no Null Space

Explanation:

The first step is to create an augmented matrix having a column of zeros.

The next step is to get this into RREF.

We can simplify to

This tells us the following.

Now we need to write this as a linear combination.

The null space is then

### Example Question #2 : Range And Null Space Of A Matrix

Find a basis for the range space of the transformation given by the matrix .

Explanation:

We can find a basis for 's range space first by finding a basis for the column space of its reduced row echelon form.

Using a calculator or row reduction, we obtain

for the reduced row echelon form.

The fourth column in this matrix can be seen by inspection to be a linear combination of the other three columns, so it is not included in our basis. Hence the first three columns form a basis for the column space of the reduced row echelon form of , and therefore the first three columns of  form a basis for its range space.

.

### Example Question #3 : Range And Null Space Of A Matrix

Find a basis for the range space of the transformation given by the matrix .

Explanation:

We can find a basis for 's range space first by finding a basis for the column space of its reduced row echelon form.

Using a calculator or row reduction, we obtain

for the reduced row echelon form.

The second column in this matrix can be seen by inspection to be a linear combination of the first column, so it is not included in our basis for . Hence the first and the third columns form a basis for the column space of , and therefore the first and the third columns of  form a basis for the range space of .

### Example Question #4 : Range And Null Space Of A Matrix

Find a basis for the range space of the transformation given by the matrix .

Explanation:

We can find a basis for 's range space first by finding a basis for the column space of its reduced row echelon form.

Using a calculator or row reduction, we obtain

for the reduced row echelon form.

The fourth column in this matrix can be seen by inspection to be a linear combination of the first three columns, so it is not included in our basis for . Hence the first three columns form a basis for the column space of , and therefore the first three columns of  form a basis for the range space of .

### Example Question #5 : Range And Null Space Of A Matrix

Find a basis for the null space of the matrix .

Explanation:

The null space of the matrix  is the set of solutions to the equation

.

We can solve the above system by row reducing  using either row reduction, or a calculator to find its reduced row echelon form. After that, our system becomes

Hence a basis for the null space is just the zero vector;

.

### Example Question #6 : Range And Null Space Of A Matrix

Find the null space of the matrix operator.

Explanation:

The null space of the operator is the set of solutions to the equation

.

We can solve the above system by row reducing our matrix using either row reduction, or a calculator to find its reduced row echelon form. After that, our system becomes

Hence the null space consists of only the zero vector.

### Example Question #7 : Range And Null Space Of A Matrix

Find the null space of the matrix .

Explanation:

The null space of the matrix  is the set of solutions to the equation

.

We can solve the above system by row reducing  using either row reduction, or a calculator to find its reduced row echelon form. After that, our system becomes

Multiplying this vector by gets rid of the fraction, and does not affect our answer, since there is an arbitrary constant behind it.

Hence the null space consists of all vectors spanned by ;

.

### Example Question #8 : Range And Null Space Of A Matrix

Find the null space of the matrix .

Explanation:

The null space of the matrix  is the set of solutions to the equation

.

We can solve the above system by row reducing  using either row reduction, or a calculator to find its reduced row echelon form. After that, our system becomes

Hence the null space consists of all vectors spanned by ;

.

### Example Question #9 : Range And Null Space Of A Matrix

What is the largest possible rank of a  matrix?

Explanation:

The rank is equal to the dimension of the row space and the column space (both spaces always have the same dimension). This matrix has three rows and five columns, which means the largest possible number of vectors in a basis for the row space of a  matrix is , so this is the largest possible rank.

### Example Question #1 : Range And Null Space Of A Matrix

What is the smallest possible nullity of a  matrix?

Explanation:

According to the Rank + Nullity Theorem,

Since the matrix has  columns, we can rearrange the equation to get

So to make the nullity as small as possible, we need to make the rank as large as possible.

The rank is equal to the dimension of the row space and the column space (both spaces always have the same dimension). This matrix has three rows and five columns, which means the largest possible number of vectors in a basis for the row space of a  matrix is , so this is the largest possible rank.

Hence the smallest possible nullity is .

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