Question

Check the following vector linearly dependent x1 = (1, 2, 1) x2 = (2, 1, 4)
x3 = (4, 5, 6).

Answer 1

$\textsf{Concept:}$

$\textsf{Three vectors v_1,\;v_2,\;v_3,\; are said to be linearly dependent if }$$\textsf{ any one of the three vectors can be written as a linear combination of other two vectors}$

$\textsf{Now,}$

$\mathsf{2\;x_1+1\;x_2}$

$\mathsf{2(1,2,1)+1(2,1,4)}$

$\mathsf{=(2,4,2)+(2,1,4)}$

$\mathsf{=(4,5,6)}$

$\mathsf{=x_3}$

$\implies\textsf{(4,5,6) is a linear combination of other two vectors}$

$\textsf{Hence, the vectors }\mathsf{x_1,\;x_2,\;x_3}\textsf{ are linearly independent}$

Answer 2

Answer:  The given vectors are linearly dependent.

Step-by-step explanation:  We are given to check whether the following vectors are linearly dependent :

$x_1= (1,2,1),~~x_2=(2,1,4),~~x_3=(4,5,6).$

We know that n vectors in an n-dimensional space are linearly dependent if their determinant is zero.

We have, for the given vectors

$det\left[\begin{array}{ccc}1&2&1\\2&1&4\\4&5&6\end{array}\right] \\\\\\=1(1\times6-4\times5)+2(4\times4-2\times6)+1(2\times5-4\times1)\\\\=1(6-20)+2(16-12)+1(10-4)\\\\=-14+8+6\\\\=0.$

Thus, the given vectors are linearly dependent.