Science:Math Exam Resources/Courses/MATH307/April 2009/Question 03 (b)

MATH307 April 2009

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Question 03 (b)
Let $v_{1}={\begin{bmatrix}1\\0\\1\\0\end{bmatrix}}$, $v_{2}={\begin{bmatrix}1\\2\\0\\1\end{bmatrix}}$,$v_{3}={\begin{bmatrix}1\\1\\-1\\-1\end{bmatrix}}$, (b) Explain how you could use MATLAB/Octave to determine whether $v_{1},v_{2}$ and $v_{3}$ are linearly dependent. What output would tell you that the vectors are independent?

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
Suppose we formed a matrix using the given three vectors. What can we learn from the reduced row echelon form of this matrix?

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Solution

Step 1: Enter the three vectors $v_{1},v_{2}$ and $v_{3}$. The code is: v1 = [1, 0, 1, 0]’; v2 = [1, 2, 0, 1]’; v3 = [1, 1, -1, -1]’; Notice that ’ is to transfer the vector to its transpose. Step 2: Combine all the three vectors into one matrix as V. The code: V = [v1, v2, v3]; Step 3: Find the reduced row echelon form of the matrix using rref() command. The code is: A = rref(V) Step 4, based on the reduced row echelon form we have found, determine if the these vectors are linear independent. The vectors are linear independent if and only if the reduced row echelon form of the matrix is ${\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\0&0&0\end{bmatrix}}$

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