Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 06

MATH152 April 2013

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Question A 06
Is it possible for three vectors in $\mathbb {R} ^{2}$ to be linearly independent? Briefly justify your answer.

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
Try to formulate this in term of a linear system of equations.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. Suppose we have three vectors over $\mathbb {R} ^{2}$ say ${\begin{bmatrix}a\\b\end{bmatrix}}\quad {\begin{bmatrix}c\\d\end{bmatrix}}\quad {\begin{bmatrix}e\\f\end{bmatrix}}$ for fixed a,b,c,d,e,f. These vectors are linearly independent if we can write $x{\begin{bmatrix}a\\b\end{bmatrix}}+y{\begin{bmatrix}c\\d\end{bmatrix}}+z{\begin{bmatrix}e\\f\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}$ and have x=y=z=0 be the only solution. We can write this as a system of linear equations given by: ${\begin{aligned}xa+yc+ze&=0\\xb+yd+zf&=0\end{aligned}}$ This is a system of two equations and three unknowns (given by x,y, and z). This always has a nontrivial solution (i.e. for any value of z, I can always find an x and y that solve this) and therefore x=y=z=0 is not the only solution. The answer then is that 3 vectors are always linearly dependent in $\mathbb {R} ^{2}$

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