## Linear Algebra and Its Applications, Exercise 2.3.4

Exercise 2.3.4. Suppose the vectors $v_1$, $v_2$, and $v_3$ are linearly independent. (The book says linearly dependent, but I believe this is a typo.) Are the vectors $w_1 = v_1 + v_2$, $w_2 = v_1 + v_3$, and $w_3 = v_2 + v_3$ also linearly independent?

Answer: Consider the linear combination of $w_1$, $w_2$, and $w_3$ with weights $c_1$, $c_2$, and $c_3$. We have $c_1w_1 + c_2w_2 + c_3w_3 = c_1(v_1 + v_2) + c_2(v_1 + v_3) + c_3(v_2 + v_3)$ $= (c_1+c_2)v_1 + (c_1+c_3)v_2 + (c_2+c_3)v_1$

Since the vectors $v_1$, $v_2$, and $v_3$ are linearly independent the above expression can be zero only if $c_1 + c_2 = 0$, $c_1 + c_3 = 0$, and $c_2 + c_3 = 0$.

We can express this as the following system of equations $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcl}c_1&+&c_2&&&=&0 \\ c_1&&&+&c_3&=&0 \\ &&c_2&+&c_3&=&0 \end{array}$

and can solve it via elimination. We first subtract the first equation from the second to obtain the following system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcl}c_1&+&c_2&&&=&0 \\ &&-c_2&+&c_3&=&0 \\ &&c_2&+&c_3&=&0 \end{array}$

and then add the second equation to the third to obtain the final system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcl}c_1&+&c_2&&&=&0 \\ &&-c_2&+&c_3&=&0 \\ &&&&2c_3&=&0 \end{array}$

Solving for the weights we have $2c_3 = 0$ or $c_3 = 0$, $-c_2 + 0 = 0$ or $c_2 = 0$, and $c_1 + 0 = 0$ or $c_1 = 0$.

Since $c_1w_1 + c_2w_2 + c_3w_3 = 0$ only if $c_1 = c_2 = c_3 = 0$ we see that $w_1$, $w_2$, and $w_3$ are linearly independent.

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 .

This entry was posted in linear algebra. Bookmark the permalink.