Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 24

MATH152 April 2015

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Question A 24
Write $\mathbf {b} =[-5,11,18]$ as a linear combination $\mathbf {a} _{1}=[1,2,3]$ and $\mathbf {a} _{2}=[3,-1,-2]$ or show it cannot be done.

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
We want to find $x,y$ such that $x{\begin{bmatrix}1\\2\\3\end{bmatrix}}+y{\begin{bmatrix}3\\-1\\-2\end{bmatrix}}={\begin{bmatrix}-5\\11\\18\end{bmatrix}}.$ Try writing this vector equation as a system of linear equations and solving it using your favourite method. (The solution presented here will use row reduction.)

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. We want to find $x,y$ such that $x{\begin{bmatrix}1\\2\\3\end{bmatrix}}+y{\begin{bmatrix}3\\-1\\-2\end{bmatrix}}={\begin{bmatrix}-5\\11\\18\end{bmatrix}}.$ We rewrite this vector equation as the augmented matrix of a linear system and apply row reduction: $\left[{\begin{array}{cc|c}1&3&-5\\2&-1&11\\3&-2&18\end{array}}\right].$ We introduce zeroes below the first pivot by adding appropriate multiples of the first row to the second and third rows (what are these multiples?): $\left[{\begin{array}{cc|c}1&3&-5\\0&-7&21\\0&-11&33\end{array}}\right].$ We then divide the second and third rows by -7 and -11, respectively: $\left[{\begin{array}{cc|c}1&3&-5\\0&1&-3\\0&1&-3\end{array}}\right].$ This is implies that $y=-3$; substituting this into the first equation then yields $x=4$. It follows that $4{\begin{bmatrix}1\\2\\3\end{bmatrix}}-3{\begin{bmatrix}3\\-1\\-2\end{bmatrix}}={\begin{bmatrix}-5\\11\\18\end{bmatrix}},$ i.e., $\color {blue}\mathbf {b} =4\mathbf {a} _{1}-3\mathbf {a} _{2}.$