Gaussian Elimination: 4 equations, 8 unknowns

Question

I need to find the solution set of all b=(b1,b2,b3,b4) element R^4 for this:

I know how Gaussian elimination works, but how do I find all possible solutions of all the b's?

Answer 1

Look at the matrix $A$:

$$A = \begin{pmatrix} 1 & -4 & 5 & 1 \\ 1 &-2 & 3 & 1\\ 7 & -11 & 18 & 7 \\ -2 & 3 & -5 &-2 \end{pmatrix}.$$

Notice how the first and last columns are the same? This means that the matrix is singular; in other words, its rank is less than its dimension. In fact, if you perform Gaussian elimination, you will find that its rank is 2 (try it!).

What does this mean?

Well, consider $A$ as a linear transformation. Since its rank is 2, and its dimension is 4, then the dimension of its null space is also 2. That means that when you apply this linear transformation to vectors in $\mathbb{R}^4$, your result will either be zero, or it will lie in a 2-dimensional subspace of $\mathbb{R}^4$. This means that $b$ is either a vector of zeros, or it lies in this subspace. Your task is to find a basis for this subspace, and then $b$ is any linear combination of those basis vectors. Since the subspace is 2-dimensional, you should find two linearly independent vectors in $\mathbb{R}^4$ forming the basis.