Question

Test whether the set {(1,1,1,),(1,1,0),(1,0,1)} of vectors is Linearly independent or

dependent​

Answer 1

Answer:

The set of vectors {(1,1,1,),(1,1,0),(1,0,1)} are Linearly independent

Step-by-step explanation:

Tip:

• The dependency of vectors can be check by the determinant of the vectors.
• If the determinant is 0 then vectors are dependent otherwise independent.

Step1 of 1:

• The matrix formed is given by

     $\left[\begin{array}{ccc}1&1&1\\1&1&0\\1&0&1\end{array}\right]$

• Determinant of matrix is given by

       $D=1(1-0)-1(1-0)+1(-1)\\D=-1$

• Since the determinant is non zero, the vectors are independent.

Answer 2

The set {(1,1,1,) , (1,1,0) , (1,0,1)} of vectors is Linearly independent

Given :

The set {(1,1,1,) , (1,1,0) , (1,0,1)} of vectors

To find :

Test the set of vectors is Linearly independent or dependent

Concept :

If the value of the determinant of the matrix formed by the given set of vectors is non zero , then the given set of vectors is Linearly independent. Otherwise it is Linearly dependent

Solution :

Step 1 of 4 :

Write down the given set of vectors

Here the given set of vectors are {(1,1,1,) , (1,1,0) , (1,0,1)}

Step 2 of 4 :

Form the matrix with given set of vectors

Let A be the matrix form with given set of vectors

Then

$\displaystyle \sf A = \begin{pmatrix} 1 & 1 & 1\\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}$

Step 3 of 4 :

Find value of the determinant

The value of the determinant A

= det A

$\displaystyle \sf = \begin{vmatrix} 1 & 1 & 1\\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{vmatrix}$

$\displaystyle \sf = 1(1 - 0) - 1(1 - 0) + 1(0 - 1)$

$\displaystyle \sf = 1 - 1 - 1$

$\displaystyle \sf = - 1 \ne \: 0$

Step 4 of 4 :

Test the set of vectors is Linearly independent or dependent

Since the value of the determinant of the matrix formed by the given set of vectors is non zero

Hence the given set of vectors is Linearly independent

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