## Linear Algebra and Its Applications, Exercise 2.2.6

Exercise 2.2.6. Consider the following system of linear equations: $\begin{bmatrix} 1&2&2 \\ 2&4&5 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$

Find the general solution expressed as the sum of a particular solution to $Ax = b$ and the solution to $Ax = 0$.

Answer: We perform elimination by subtracting 2 times the first row from the second row: $\begin{bmatrix} 1&2&2 \\ 2&4&5 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&2&2 \\ 0&0&1 \end{bmatrix}$

This completes elimination. The matrix $U$ has pivots in columns 1 and 3, so that the basic variables are $u$ and $w$ and the free variable is $v$.

We can replace the system $Ax = 0$ with the new system $Ux = 0$: $\begin{bmatrix} 1&2&2 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

From the second equation we have $w = 0$. Substituting into the first equation gives us $u +2v + 2w = u + 2v = 0$ or $u = -2v$. The general solution to $Ax = 0$ is thus $x = \begin{bmatrix} -2v \\ v \\ 0 \end{bmatrix} = v \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}$

We now consider the inhomogeneous system $Ax = b$ or $\begin{bmatrix} 1&2&2 \\ 2&4&5 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$

The elimination sequence from above produces the system $Ux = c$ or $\begin{bmatrix} 1&2&2 \\ 0&0&1 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

From the second equation we have $w = 2$. Substituting into the first equation and setting the free variable $v = 0$ produces $u + 2v + 2w = u + 0 + 4 = 1$ or $u = -3$. The particular solution is thus $x = (-3, 0, 2)$.

We can combine the particular solution to this system with the general solution to $Ax = 0$ to produce the general solution $x$ for the system $Ax = b$: $x = \begin{bmatrix} -3 \\ 0 \\ 2 \end{bmatrix} + v \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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### 2 Responses to Linear Algebra and Its Applications, Exercise 2.2.6

1. Nate says:

you solved the particular solution right but copied it wrong in your total solution

• hecker says:

Thanks for catching this! I’ve updated the post to fix it.