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Determine if the columns of the matrix form a linearly independent set. Justify your answer.

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Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.)

**A.**If A is the given matrix, then the augmented matrix ( ) represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A form a linearly independent set.

**B.**If A is the given matrix, then the augmented matrix ( ) represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has more than one solution. Therefore, the columns of A do not form a linearly independent set.

**C.**If A is the given matrix, then the augmented matrix ( ) represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.

**D.**If A is the given matrix, then the augmented matrix ( ) represents the equation Ax = 0. The reduced echelon form of this matrix indicates that Ax = 0 has only the trivial solution. Therefore, the columns of A form a linearly independent set.

♦ Relevant knowledge

A set is described as linearly independent when all the vectors cannot be described as the result of a linear combination of different vectors. Additionally, the amount of the independent linear vectors is known as the size of the space created by the set.

Answer:Data provided:

It is necessary to determine if the columns in the matrix form a linearly-independent set.

To determine if the columns in the matrix form a linearly-independent set, we must solve the equation given below.

The augmented matrix can be written as follows:

Let’s now reduce the matrix to the row-echelon reduced form.

We get the following results from by interchanging row1 with row3.

We get – after performing

On performing , we get –

On performing , we get –

On performing , we get –

On performing , we get –

On performing , we get –

On performing , we get –

On performing , we get –

On performing , we get –

On performing , we get –

The matrix above is in reduced row-echelon format.

The equation has a trivial solution.

If A is the given matrix, then the augmented corresponds to the equation

This matrix’s reduced row-echelon form indicates that Ax=0 only has the trivial solution.

The columns of A are therefore a linearly dependent set.

Therefore, the best option is (D).