Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 03

MATH152 April 2010

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Question A 03
For questions A1-A3 below, $\mathbf {x}$ and $\mathbf {y}$ are the given vectors below. $\mathbf {x} =[1,-1,-1],\quad \mathbf {y} =[1,1,2],$ A3: Are $\mathbf {x}$ and $\mathbf {y}$ linearly independent? Justify briefly.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Two vectors x and y are linearly dependent if one is a multiple of the other.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From the hint we see that two vectors x and y are linearly dependent if they are multiples of one another. This would mean that we can write, $\mathbf {x} =k\mathbf {y}$ for some constant k. This is equivalent to ${\begin{bmatrix}1\\-1\\-1\end{bmatrix}}=k{\begin{bmatrix}1\\1\\2\end{bmatrix}}$ or ${\begin{aligned}1&=k\\-1&=k\\-1&=2k\end{aligned}}$ The first equation tells us k=1 but the other two equations then say that 1=-1 and 2=-1 which are both false and therefore there does not exist a unique k such that x=ky and thus x and y are not multiples of one another so they are not linearly dependent.

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