Linear Algebra and Its Applications, exercise 1.2.5

Exercise 1.2.5. The following three equations:

$\setlength\arraycolsep{0.2em}\begin{array}{rcl} t&=&0 \\ z&=&0 \\ x + y + z + t&=&1 \end{array}$

correspond to planes in 4-space (or space-time). Find two points on the line that is the intersection of the planes represented by the equations.

Answer: We substitute the values of t and z from the first two equations into the third:

$x + y + z + t = 1 \Rightarrow x + y = 1$

Values of x and y satisfying this equation include x = 1, y = 0 and x = 0, y = 1. Two points satisfying all three equations are thus (1, 0, 0, 0) and (0, 1, 0, 0).

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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1 Response to Linear Algebra and Its Applications, exercise 1.2.5

Trevor Stern says:

August 1, 2022 at 8:50 pm

Thank you very much for posting these exercises in Linear Algebra and It’s Applications. You are a godsend!