Please help me with this Linear Algebra question which invol

Please help me with this Linear Algebra question which involves finding a basis for six vectors. Please be DETAILED. If you cannot be detailed, don\'t answer the question. I will rate the answer based on the explanation and work shown. I\'m having trouble with this material and need to understand it.

Thank you.

Solution

Step 1: Set up a homogeneous system of equations

The set S = {v₁, v₂, v₃, v₄} of vectors in R⁶ is linearly independent if the only solution of

(*)    c₁v₁ + c₂v₂ + c₃v₃ + c₄v₄ = 0

is c₁, c₂, c₃, c₄ = 0.

In this case, the set S forms a basis for span S.

Otherwise (i.e., if a solution with at least some nonzero values exists), S is linearly dependent.

If this is the case, a subset of S can be found that forms a basis for span S.

With our vectors v₁, v₂, v₃, v₄, (*) becomes:

Rearranging the left hand side yields

The matrix equation above is equivalent to the following homogeneous system of equations

We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form to determine whether the system has

can be transformed by a sequence of elementary row operations to the matrix

The reduced row echelon form of the coefficient matrix of the homogeneous system (**) is

which corresponds to the system

Since each column contains a leading entry (highlighted in yellow), then the system has only the trivial solution, so that the only solution of (*) is c₁, c₂, c₃, c₄ = 0.

Therefore the set S = {v₁, v₂, v₃, v₄} is linearly independent.

Consequently, the set S forms a basis for span S.