Answer:
The vectors are linearly independent
Step-by-step explanation:
These vectors can be written in a matrix form as:
[tex]\left[\begin{array}{ccc}5&-2&4\\2&-3&5\\4&5&-7\end{array}\right][/tex]
and if the matrix is invertible, the system has a unique solution, and hence, the vectors that form the matrix are linearly independent.
A matrix is invertible if it's determinant is different from zero.
Suppose A is a matrix, A is invertible if |A| ≠ 0
[tex]\left|\begin{array}{ccc}5&-2&4\\2&-3&5\\4&5&-7\end{array}\right| \\[/tex]
= [tex]5\left|\begin{array}{cc}-3&5\\5&-7\end{array}\right| - (-2)\left|\begin{array}{cc}2&5\\4&-7\end{array}\right| + 4\left|\begin{array}{cc}2&-3\\4&5\end{array}\right|[/tex]
= [tex]= 5[-7(-3) - 5(5)] + 2[2(-7) - 4(5)] + 4[5(2) - 4(-3)]\\= 5(21 - 25) + 2( -14 -20) + 4(10 + 12)\\= 5(-4) + 2(-34) + 4(21)\\= -20 - 68 + 84\\= -4\\[/tex]
Since this is not zero, we conclude that the vectors are linearly independent.
