# Linear Algebra : Linear Independence and Rank

## Example Questions

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### Example Question #1 : Linear Independence And Rank

Determine whether the following vectors in Matrix form are Linearly Independent.

The vectors are Linearly Independent

The vectors aren't Linearly Independent

The vectors are Linearly Independent

Explanation:

To figure out if the matrix is independent, we need to get the matrix into reduced echelon form. If we get the Identity Matrix, then the matrix is Linearly Independent.

Since we got the Identity Matrix, we know that the matrix is Linearly Independent.

### Example Question #2 : Linear Independence And Rank

Find the rank of the following matrix.

Explanation:

We need to get the matrix into reduced echelon form, and then count all the non all zero rows.

The rank is 2, since there are 2 non all zero rows.

### Example Question #3 : Linear Independence And Rank

Calculate the Rank of the following matrix

Explanation:

We need to put the matrix into reduced echelon form, and then count all the non-zero rows.

Since there is only 1 non-zero row, the Rank is 1.

### Example Question #4 : Linear Independence And Rank

Determine if the following matrix is linearly independent or not.

Linearly Independent

Linearly Dependent

Linearly Dependent

Explanation:

Since the matrix is , we can simply take the determinant. If the determinant is not equal to zero, it's linearly independent. Otherwise it's linearly dependent.

Since the determinant is zero, the matrix is linearly dependent.

### Example Question #5 : Linear Independence And Rank

If matrix A is a 5x8 matrix with a two-dimensional null space, what is the rank of A?

Explanation:

Given that rank A + dimensional null space of A = total number of columns, we can determine rank A = total number of columns-dimensional null space of A. Using the information given in the question we can solve for rank A:

### Example Question #6 : Linear Independence And Rank

If matrix A is a 10x12 matrix with a three-dimensional null space, what is the rank of A?

Explanation:

Given that rank A + dimensional null space of A = total number of columns, we can determine rank A = total number of columns-dimensional null space of A. Using the information given in the question we can solve for rank A:

### Example Question #7 : Linear Independence And Rank

Does the following row reduced echelon form of a matrix represent a linearly independent set?

Yes

Not enough information

No

Yes

Explanation:

The set must be linearly independent because there are no rows of all zeros. There are columns of all zeros, but columns do not tell us if the set is linearly independent or not.

### Example Question #8 : Linear Independence And Rank

In a vector space of dimension 5, can you have a linearly independent set of 3 vectors?

Yes

Not enough information

No

Yes

Explanation:

The dimension of the vector space is the maximum number of vectors in a linearly independent set. It is possible to have linearly independent sets with less vectors than the dimension.

So for this example it is possible to have linear independent sets with

1 vector, or 2 vectors, or 3 vectors, all the way up to 5 vectors.

### Example Question #9 : Linear Independence And Rank

Consider a set of 3 vectors from a 3 dimensional vector space.

Is the set linearly independent?

No

Not enough information

Yes

Not enough information

Explanation:

It depends on what the vectors are.

For example, if

Then the set is linearly independent.

However if the vectors were

then the set would be linearly dependent.

### Example Question #10 : Linear Independence And Rank

Consider a set of 3 vectors from a 2 dimensional vector space.

Is the set linearly independent?

Yes

Not enough information

No

No

Explanation:

Since the dimension of the space is 2, a linearly independent set can have at most two vectors. Since the set in consideration has 3 and 3>2, the set must be linearly dependent.

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