Linear Algebra and Its Applications, Exercise 2.2.2

Exercise 2.2.2. Consider a system of linear equations that has more unknowns than there are equations and that has no solution. Provide the smallest example of such a system you can think of.

Answer: Systems of one linear equation and two unknowns (i.e., of the form ax + by = c) always have solutions. To find a system with no solution we therefore have to look at systems of two equations and three unknowns. For example, the system

\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&0 \\ u&+&v&+&w&=&1 \end{array}

has no solution: Gaussian elimination (i.e., by subtracting the first equation from the second) would produce the contradiction 0 = 1.

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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