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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