General=Particular+Homogeneous

Solve this system. Then solve the associated homogeneous system.

Solve each system. Express the solution set using vectors. Identify a particular solution and the solution set of the homogeneous system.

• (a)  $\begin{array}{rcrcr}3x&+&6y&=&18\\ x&+&2y&=&6\end{array}$     (b)  $\begin{array}{rcrcr}x&+&y&=&1\\ x&-&y&=&-1\end{array}$     (c)  $\begin{array}{rcrcrcr}x_{1}&&&+&x_{3}&=&4\\ x_{1}&-&x_{2}&+&2x_{3}&=&5\\ 4x_{1}&-&x_{2}&+&5x_{3}&=&17\end{array}$     (d)  $\begin{array}{rcrcrcr}2a&+&b&-&c&=&2\\ 2a&&&+&c&=&3\\ a&-&b&&&=&0\end{array}$     (e)  $\begin{array}{rcrcrcrcr}x&+&2y&-&z&&&=&3\\ 2x&+&y&&&+&w&=&4\\ x&-&y&+&z&+&w&=&1\end{array}$     (f)  $\begin{array}{rcrcrcrcr}x&&&+&z&+&w&=&4\\ 2x&+&y&&&-&w&=&2\\ 3x&+&y&+&z&&&=&7\end{array}$

Solve each system, giving the solution set in vector notation. Identify a particular solution and the solution of the homogeneous system.

• (a)  $\begin{array}{rcrcrcr}2x&+&y&-&z&=&1\\ 4x&-&y&&&=&3\end{array}$     (b)  $\begin{array}{rcrcrcrcr}x&&&-&z&&&=&1\\ &&y&+&2z&-&w&=&3\\ x&+&2y&+&3z&-&w&=&7\end{array}$     (c)  $\begin{array}{rcrcrcrcr}x&-&y&+&z&&&=&0\\ &&y&&&+&w&=&0\\ 3x&-&2y&+&3z&+&w&=&0\\ &&-y&&&-&w&=&0\end{array}$     (d)  $\begin{array}{rcrcrcrcrcr}a&+&2b&+&3c&+&d&-&e&=&1\\ 3a&-&b&+&c&+&d&+&e&=&3\end{array}$

For the system

which of these can be used as the particular solution part of some general solution?

• (a)  $\begin{pmatrix}0\\ -3\\ 5\\ 0\end{pmatrix}$     (b)  $\begin{pmatrix}2\\ 1\\ 1\\ 0\end{pmatrix}$     (c)  $\begin{pmatrix}-1\\ -4\\ 8\\ -1\end{pmatrix}$

Lemma 3.7 says that we can use any particular solution for $\vec{p}$. Find, if possible, a general solution to this system

that uses the given vector as its particular solution.

• (a)  $\begin{pmatrix}0\\ 0\\ 0\\ 4\end{pmatrix}$     (b)  $\begin{pmatrix}-5\\ 1\\ -7\\ 10\end{pmatrix}$     (c)  $\begin{pmatrix}2\\ -1\\ 1\\ 1\end{pmatrix}$

One is nonsingular while the other is singular. Which is which?

• (a)  $\begin{pmatrix}1&3\\ 4&-12\end{pmatrix}$     (b)  $\begin{pmatrix}1&3\\ 4&12\end{pmatrix}$

Singular or nonsingular?

• (a)  $\begin{pmatrix}1&2\\ 1&3\end{pmatrix}$     (b)  $\begin{pmatrix}1&2\\ -3&-6\end{pmatrix}$     (c)  $\begin{pmatrix}1&2&1\\ 1&3&1\end{pmatrix}$     (d)  $\begin{pmatrix}1&2&1\\ 1&1&3\\ 3&4&7\end{pmatrix}$     (e)  $\begin{pmatrix}2&2&1\\ 1&0&5\\ -1&1&4\end{pmatrix}$

Is the given vector in the set generated by the given set?

• $\begin{pmatrix}2\\ 3\end{pmatrix},$ $\{\hskip 1.00006pt\begin{pmatrix}1\\ 4\end{pmatrix},\begin{pmatrix}1\\ 5\end{pmatrix}\hskip 1.00006pt\}$

• $\begin{pmatrix}-1\\ 0\\ 1\end{pmatrix},$ $\{\hskip 1.00006pt\begin{pmatrix}2\\ 1\\ 0\end{pmatrix},\begin{pmatrix}1\\ 0\\ 1\end{pmatrix}\hskip 1.00006pt\}$

• $\begin{pmatrix}1\\ 3\\ 0\end{pmatrix},$ $\{\hskip 1.00006pt\begin{pmatrix}1\\ 0\\ 4\end{pmatrix},\begin{pmatrix}2\\ 1\\ 5\end{pmatrix},\begin{pmatrix}3\\ 3\\ 0\end{pmatrix},\begin{pmatrix}4\\ 2\\ 1\end{pmatrix}\hskip 1.00006pt\}$

• $\begin{pmatrix}1\\ 0\\ 1\\ 1\end{pmatrix},$ $\{\hskip 1.00006pt\begin{pmatrix}2\\ 1\\ 0\\ 1\end{pmatrix},\begin{pmatrix}3\\ 0\\ 0\\ 2\end{pmatrix}\hskip 1.00006pt\}$

Prove that any linear system with a nonsingular matrix of coefficients has a solution, and that the solution is unique.

In the proof of Lemma 3.6, what happens if there are no non-$0=0$ equations?

Prove that if $\vec{s}$ and $\vec{t}$ satisfy a homogeneous system then so do these vectors.

• (a)  $\vec{s}+\vec{t}$     (b)  $3\vec{s}$     (c)  $k\vec{s}+m\vec{t}$ for $k,m\in\mathbb{R}$

What’s wrong with this argument: “These three show that if a homogeneous system has one solution then it has many solutions — any multiple of a solution is another solution, and any sum of solutions is a solution also — so there are no homogeneous systems with exactly one solution.”?

Prove that if a system with only rational coefficients and constants has a solution then it has at least one all-rational solution. Must it have infinitely many?