Below is a clear definition of of Linear Independence with examples for illustrations

If at least one of the three equations can be
described in terms of the other equations, then the system is said to be linearly dependent. If there is no way
at all to express at least one of the equations as a linear combination of the
other equations, then the system is referred to as linearly independent

Suppose that you have the following equations;

x + y +z = 0

3x – 2y- z = 0

2x + 3y – z= 0

Testing
for Linear Independence

We use the determinant of the matrix to test for
linear independence. If the resulting determinant is zero then the equations
are said to be linearly dependent. If not then there is linear independence.

Example
1

Suppose we have the following system of two
equations

3x + 2y = 0

4x + 5y = 0

Our Matrix will be

Calculating the matrix= (3*5)-(2*4) = 9

Since the determinant of 9 is greater than 0 then
the equations are linearly independent.

Example
2

x + 2y = 0

2x + 4y = 0

Our Matrix will be

Calculating the matrix= (1*4)-(2*2) = 0

Since the resulting determinant is equal to 0 then the equations are linearly dependent.

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