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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).