## Linear Algebra and Its Applications, exercise 1.3.3

Exercise 1.3.3. Given the following system of equations: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcrcl}2u&-&v&&&&&=&0 \\ -u&+&2v&-&w&&&=&0 \\ &&-v&+&2w&-&z&=&0 \\ &&&&-w&+&2z&=&5 \end{array}$

find a solution to the system, and give the pivots. You can use a matrix to represent the system (including the right-hand side).

Answer: We can represent the system as the following matrix: $\left[ \begin{array}{rrrrr} 2&-1&0&0&0 \\ -1&2&-1&0&0 \\ 0&-1&2&-1&0 \\ 0&0&-1&2&5 \end{array} \right]$

The first pivot is 2. The first step in elimination gives the following matrix: $\left[ \begin{array}{rrrrr} 2&-1&0&0&0 \\ 0&\frac{3}{2}&-1&0&0 \\ 0&-1&2&-1&0 \\ 0&0&-1&2&5 \end{array} \right]$

The second pivot is 3/2. The second step in elimination gives the following: $\left[ \begin{array}{rrrrr} 2&-1&0&0&0 \\ 0&\frac{3}{2}&-1&0&0 \\ 0&0&\frac{4}{3}&-1&0 \\ 0&0&-1&2&5 \end{array} \right]$

The third pivot is 4/3. The third step in elimination gives the following: $\left[ \begin{array}{rrrrr} 2&-1&0&0&0 \\ 0&\frac{3}{2}&-1&0&0 \\ 0&0&\frac{4}{3}&-1&0 \\ 0&0&0&\frac{5}{4}&5 \end{array} \right]$

Now that the system is in trangular form we re-express it using u, v, w, and z: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcrcl}2u&-&v&&&&&=&0 \\ &&\frac{3}{2}v&-&w&&&=&0 \\ &&&&\frac{4}{3}w&-&z&=&0 \\ &&&&&&\frac{5}{4}z&=&5 \end{array}$

and then back-substitute, starting with solving for z: $\begin{array}{rcrcr}\frac{5}{4}z = 5&\Rightarrow&z = 4&& \\ \frac{4}{3}w - z = 0&\Rightarrow&\frac{4}{3}w = 4&\Rightarrow&w = 3 \\ \frac{3}{2}v - w = 0&\Rightarrow&\frac{3}{2}v = 3&\Rightarrow&v = 2 \\ 2u - v = 0&\Rightarrow&2u = 2&\Rightarrow&u = 1\end{array}$

So the solution is u = 1, v = 2, w = 3, z = 4.

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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