## 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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