## Linear Algebra and Its Applications, Exercise 1.5.18

Exercise 1.5.18. We have the following systems of linear equations: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}&&v&-&w&=&2 \\ u&-&v&&&=&2 \\ u&&&-&w&=&2 \end{array}$ and $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}&&v&-&w&=&0 \\ u&-&v&&&=&0 \\ u&&&-&w&=&0 \end{array}$ and $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}&&v&+&w&=&1 \\ u&+&v&&&=&1 \\ u&&&+&w&=&1 \end{array}$

Which of the systems are singular, and which nonsingular? Which have no solutions? One solution? An infinite number of solutions?

Answer: For the first system we can add the first and second equations to obtain the following system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}&&v&-&w&=&2 \\ u&-&v&&&=&2 \\ u&&&-&w&=&2 \end{array} \rightarrow \setlength\arraycolsep{0.2em}\begin{array}{rcrcr}u&-&w&=&4 \\ u&-&w&=&2 \end{array}$

This system is singular and has no solution.

For the second system we can again add the first and second equations to obtain an equivalent system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}&&v&-&w&=&0 \\ u&-&v&&&=&0 \\ u&&&-&w&=&0 \end{array} \rightarrow \setlength\arraycolsep{0.2em}\begin{array}{rcrcr}u&-&w&=&0 \\ u&-&w&=&0 \end{array} \rightarrow u = w$

This system is also singular, and has infinitely many solutions.

For the third system we can subtract the first equation from the second: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}&&v&+&w&=&1 \\ u&+&v&&&=&1 \\ u&&&+&w&=&1 \end{array} \rightarrow \setlength\arraycolsep{0.2em}\begin{array}{rcrcr}u&-&w&=&0 \\ u&+&w&=&1 \end{array} \rightarrow u = \frac{1}{2}, v = \frac{1}{2}, w = \frac{1}{2}$

This system is nonsingular and has one solution.

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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