## 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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