## Linear Algebra and Its Applications, Exercise 2.2.3

Exercise 2.2.3. Consider the system of linear equations represented by the following matrix: $A = \begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 1&2&0&1 \end{bmatrix}$

Find the factorization $A = LU$, a set of basic variables, a set of free variables, a general solution to $Ax = 0$ (expressed as a linear combination as in equation (1) on page 73), and the rank of $A$.

Answer: We perform elimination on $A$ by subtracting the first row from the third (i.e., using the multiplier $l_{31} = 1$): $\begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 1&2&0&1 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 0&0&0&0 \end{bmatrix}$

This completes elimination, and leaves us with the factorization $U = \begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 0&0&0&0 \end{bmatrix} \qquad L = \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 1&0&1 \end{bmatrix}$

so that $LU = \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 1&0&1 \end{bmatrix} \begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 0&0&0&0 \end{bmatrix} = \begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 1&2&0&1 \end{bmatrix} = A$

We now solve for $Ax = \begin{bmatrix} 1&2&0&1 \\ 0&1&1&0 \\ 1&2&0&1 \end{bmatrix} \begin{bmatrix} u \\ v \\ w \\ y \end{bmatrix} = 0$

The pivots in $U$ are in columns 1 and 2, so the basic variables are $u$ and $v$ and the free variables are $w$ and $y$. The rank of $A$ is 2, the number of basic variables (or pivots).

From the second row of $A$ we have $v + w = 0$ and thus $v = -w$. From the first row of $A$ we have $u + 2v + y = 0$ and can substitute for $v$ to obtain $u - 2w + y = 0$ or $u = 2w - y$. We then have $x = \begin{bmatrix} 2w - y \\ -w \\ w \\ y \end{bmatrix} = w \begin{bmatrix} 2 \\ -1 \\ 1 \\ 0 \end{bmatrix} + y \begin{bmatrix} -1 \\ 0 \\ 0 \\ 1 \end{bmatrix}$

UPDATE: Corrected the final equation (had $w$ multiplying the second vector instead of $y$).

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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