Finding a Basis in Vector Spaces
<p>To determine a basis for the vector space V = Span(v₁, v₂, v₃, v₄, v₅), follow these steps:</p>
<p>1. Form a matrix A with the vectors v₁, v₂, v₃, v₄, and v₅ as columns:</p>
<p>A = \begin{bmatrix} -2 & -1 & 0 & 2 & 1 \\ 3 & 4 & 1 & -8 & 1 \\ 4 & 0 & 4 & 0 & 1 \\ 2 & 2 & -2 & 0 & 1 \end{bmatrix}</p>
<p>2. Reduce the matrix A to its row echelon form (REF) or reduced row echelon form (RREF) using Gaussian elimination.</p>
<p>3. Identify the pivot columns in the REF or RREF; these correspond to the vectors that form the basis of V.</p>
<p>4. The selected vectors can then be written as a linear combination of the original vectors v₁, v₂, v₃, v₄, v₅.</p>
<p>For part (b), list all possible bases formed by linear combinations of the remaining vectors based on the identified basis from part (a).</p>
