Question

Question: Determine whether the statements that follow are true or false, and justify your answer.

48.If vector$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{w}\mathbf{}}$ is a linear combination of $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{u}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$ , then $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{u}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{w}\mathbf{}}$ must be a linear combination of$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{u}}$ and$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{u}}\mathbf{+}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$ .

Short answer

The statement, “If vector $\stackrel{\rightharpoonup }{w}$ is a linear combination of $\stackrel{\rightharpoonup }{u}and\stackrel{\rightharpoonup }{v}$ , then localid="1659960886081" $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}$ must be a linear combination of $\stackrel{\rightharpoonup }{u}$and $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}$.” is false.

Step-by-step solution

Consider the condition

Let the vector$\stackrel{\rightharpoonup }{w}$ with linear combination of $\stackrel{\rightharpoonup }{u}and\stackrel{\rightharpoonup }{v}$ be, $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}$, where and are scalar.

Then, the vector, $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}$becomes,

$$\begin{gathered} \stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}=\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{xu}+y\stackrel{\rightharpoonup }{v} \\ =\stackrel{\rightharpoonup }{u}(1+x)+\stackrel{\rightharpoonup }{v}(1+y) \end{gathered}$$

Then, the vector, $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}$ is a linear combination of $\stackrel{\rightharpoonup }{u}and\stackrel{\rightharpoonup }{v}$.

Conclusion

Hence, the vector $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}+\stackrel{\rightharpoonup }{w}$ is a linear combination of $\stackrel{\rightharpoonup }{u}and\stackrel{\rightharpoonup }{v}$ and not $\stackrel{\rightharpoonup }{u}$ and $\stackrel{\rightharpoonup }{u}+\stackrel{\rightharpoonup }{v}$.