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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