For each of the following inhomogeneous systems, determine...

For each of the following inhomogeneous systems, determine whether the right-hand side lies in the image of the coefficient matrix, and, if so, write out the general solution, clearly identifying the particular solution and the kernel element. (a) $\left(\begin{array}{ll}1 & -1 \\ 3 & -3\end{array}\right) \mathrm{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)$, (b) $\left(\begin{array}{rrr}2 & 1 & 4 \\ -1 & 2 & 1\end{array}\right) \mathbf{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)$, (c) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 1 \\ 1 & -2 & 2\end{array}\right) \mathbf{x}=\left(\begin{array}{l}0 \\ 3 \\ 3\end{array}\right)$. (d) $\left(\begin{array}{rr}-2 & 1 \\ -2 & 3 \\ 3 & -5\end{array}\right) \mathbf{x}=\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right)$, (e) $\left(\begin{array}{rrrr}-1 & 3 & 0 & 2 \\ 2 & -6 & 1 & -1 \\ -3 & 9 & -2 & 0\end{array}\right) \mathbf{x}=\left(\begin{array}{r}2 \\ -2 \\ 2\end{array}\right)$.

For each of the following inhomogeneous systems, determine whether the right-hand side lies in the image of the coefficient matrix, and, if so, write out the general solution, clearly identifying the particular solution and the kernel element.
(a) $\left(\begin{array}{ll}1 & -1 \\ 3 & -3\end{array}\right) \mathrm{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(b) $\left(\begin{array}{rrr}2 & 1 & 4 \\ -1 & 2 & 1\end{array}\right) \mathbf{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 1 \\ 1 & -2 & 2\end{array}\right) \mathbf{x}=\left(\begin{array}{l}0 \\ 3 \\ 3\end{array}\right)$.
(d) $\left(\begin{array}{rr}-2 & 1 \\ -2 & 3 \\ 3 & -5\end{array}\right) \mathbf{x}=\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}-1 & 3 & 0 & 2 \\ 2 & -6 & 1 & -1 \\ -3 & 9 & -2 & 0\end{array}\right) \mathbf{x}=\left(\begin{array}{r}2 \\ -2 \\ 2\end{array}\right)$.

Key Concepts

General Solution Decomposition

Any solution to the inhomogeneous system Ax = b (when it is consistent) can be written in the form x = x_p + x_h, where x_p is a particular solution to Ax = b and x_h is any solution from the kernel of A (Ax = 0). This decomposition emphasizes the structure of the solution space as an affine subspace parallel to the kernel of the matrix.

Kernel of a Matrix (Null Space)

The kernel, or null space, of a matrix A is the set of all vectors x such that Ax = 0. This concept is crucial because once a particular solution to the inhomogeneous system is found, all solutions can be expressed as the sum of this particular solution and any element from the kernel.

Inhomogeneous Linear Systems

An inhomogeneous linear system is a set of linear equations of the form Ax = b where A is a matrix and b is a nonzero vector. The key issue is determining if b is in the image of A, meaning there exists at least one solution x such that Ax = b. Assessing the consistency of such systems involves comparing the rank of A with the augmented matrix [A, b].

Image of a Matrix (Column Space)

The image, or column space, of a matrix A consists of all vectors b for which there exists an x solving Ax = b. When solving inhomogeneous systems, one must verify that the right-hand side vector b lies in this column space, as this is a necessary condition for the existence of solutions.

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