Question

Question: Apply the Linear dependence and Linear Independence in vector definitions and show that the given vectors are Linearly dependant or independent vectors in R⁴.

V₁ = (1, 3, -1, 4)^T, V₂= (3, 8, -5, 7)^T, V₃= (2, 9, 4, 23)^T.

Answer 1

Solution:

Given, V₁ = (1, 3, -1, 4)^T, V₂= (3, 8, -5, 7)^T, V₃= (2, 9, 4, 23)^T

We have,

a(1, 3, -1, 4)^T +b(3, 8, -5, 7)^T +c(2, 9, 4, 23)^T=(0 0 0 0)^T

Or we can write as follows:

$a+3b+2c=0\ ...(i) \\ 3a+8b+9c=0\ ...(ii) \\ -a-5b+4c=0\ ...(iii) \\ 4a+7b+23c=0\ ...(iv)$

On solving (i), (ii) and (iii), we get

$a=-11c,\:b=3c \ ...(v)$

Put (v) in (iv).

$4a+7b+23c=0 \\ \Rightarrow 4(-11c)+7(3c)+23c=0 \\ \Rightarrow -44c+21c+23c=0 \\ \Rightarrow 0=0$

It means it is true for all c.

Thus, we don't have any of a,b,c non-zero.

So, given vectors are linearly dependent.