Consider the curve $a \subset \mathbb{R^3}$ given by the formula $$a(t) = (2t,t^2,t^3/3)$$ where $t \in \mathbb{R}$ Verify that $\{a',a''\}$ is a linearly independent set for each $t$.
Verifying linearly independent set
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linear-algebra
differential-geometry
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0what is the meaning of $$a',a''$$? the derivative with respect to $$t$$? – 2017-01-05
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0Yes, the first and second derivative. – 2017-01-05
1 Answers
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then we have $$a'(t)=(2,2t,t^2)$$ and $$a''(t)=(0,2,2t)$$ if these vectors are linear independet, then must have the equationsystem $$\alpha(2,2t,t^2)+\beta(0,2,2t)=(0,0,0)$$ has only the solution $$\alpha=0,\beta=0$$
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0Ok this is indeed what I did. Thank you for the response, I needed the confidence in my thinking. Ill tick in 2 minutes. – 2017-01-05