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

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