I'm doing some of the even problems from my textbook that don't have answers.

Given: $\vec{a}$, $\vec{b}$ and $\vec{c}$ are non-$\vec{0}$ $\in F$

  • $\vec{c}$ is not a multiple of $\vec{b}$
  • $\vec{a} =\vec{b}+4i\vec{c}$

Is $\{\vec{a},\vec{b}\}$ linearly independent or linearly dependent or neither?

My reasoning is neither because there isn't enough information to make either determination but I'm unsure. I don't know how I might might assort $c_1\vec{a} + c_2\vec{b}=\vec{0}$ from the information given so I believe it cannot be linearly dependent, but the linear independence is harder to figure out.

Since I know that the set can't be linearly dependent can I say a set of vectors is linearly independent? Is this an example of needing to use the contrapositive to solve the problem?


If $\vec{a}$ and $\vec{b}$ were linearly dependent, you would have that $\vec{a}=\lambda \vec{b}$, and then $$\vec{c}=\frac1{4i}(\vec{a}-\vec{b})=\frac{\lambda-1}{4i}\vec{b},$$ which contradicts the hypothesis that $\vec{c}$ is not a multiple of $\vec{b}$.

  • $\begingroup$ The two bullet points were two of the givens in the problem, I have to work from that information. EDIT: I SEE, I'M SORRY $\endgroup$ – jake mckenzie Oct 16 '16 at 17:50
  • $\begingroup$ That's what I did. The first equality I wrote is (equivalent to) your second bullet point, and I obtained something that contradicts your second bullet point, which means my initial assumption was incorrect and hence $\vec{a}$ and $\vec{b}$ must be linearly independent. $\endgroup$ – Arnaud D. Oct 16 '16 at 17:56

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