Linear Algebra and Its Applications, Exercise 2.2.10

Exercise 2.2.10. (a) Find all possible solutions to the following system:

$Ux = \begin{bmatrix} 1&2&3&4 \\ 0&0&1&2 \\ 0&0&0&0 \end{bmatrix} \begin{bmatrix} x_1 \\x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

(b) What are the solutions if the right side is replaced with $(a, b, 0)$?

Answer: (a) Since the pivots of $U$ are in columns 1 and 3, the basic variables are $x_1$ and $x_3$ and the free variables are $x_2$ and $x_4$.

From the second equation we have $x_3 + 2x_4 = 0$ or $x_3 = -2x_4$. Substituting the value of $x_3$ into the first equation we have $x_1 + 2x_2 - 6x_4 + 4x_4 = 0$ or $x_1 = -2x_2 + 2x_4$. The solution to $Ux = 0$ can then be expressed in terms of the free variables $x_2$ and $x_4$ as follows:

$x_{homogeneous} = x_2 \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} 2 \\ 0 \\ -2 \\ 1 \end{bmatrix}$

(b) Replacing f the right side with $(a, b, 0)$ produces the following system:

$Ux = \begin{bmatrix} 1&2&3&4 \\ 0&0&1&2 \\ 0&0&0&0 \end{bmatrix} \begin{bmatrix} x_1 \\x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} a \\ b \\ 0 \end{bmatrix}$

From the second equation we have $x_3 + 2x_4 = b$ or $x_3 = b - 2x_4$. Substituting for $x_3$ into the first equation we have $x_1 + 2x_2 + 3(b - 2x_4) + 4x_4 = a$ or $x_1 = a - 3b - 2x_2 - 2x_4$. Setting the free variables $x_2$ and $x_4$ to zero gives the particular solution

$x_{particular} = \begin{bmatrix} a - 3b \\ 0 \\ b \\ 0 \end{bmatrix}$

The general solution is then

$x = x_{particular} + x_{homogeneous}$

$= \begin{bmatrix} a-3b \\ 0 \\ b \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} 2 \\ 0 \\ -2 \\ 1 \end{bmatrix}$

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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