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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