Science:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (a)

MATH221 April 2013

  •  Q1  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8  •  Q9  •  Q10  •  Q11  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  

   Other MATH221 Exams

  •  December 2008  •  December 2009  •  December 2007  •  April 2010  •  April 2009  •  December 2011  •  April 2013  •  

Question 02 (a)
Let $A=\left[{\begin{array}{ccccc}1&2&1&3&2\\3&4&9&0&7\\2&3&5&1&8\\2&2&8&-3&5\end{array}}\right]B=\left[{\begin{array}{ccccc}1&2&1&3&2\\0&1&-3&5&-4\\0&0&0&1&-7\\0&0&0&0&0\end{array}}\right]$ The matrices A and B are related by $B=MA$ , where M is an invertible 4x4 matrix. a) Find the rank and a basis of the row space of A.

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
How does left multiplication of a matrix by a invertible matrix change the rank and row space? What are properties preserved under those changes?

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. Since the matrix A and B are related by the left multiplication of that invertible matrix M, they share many characteristic, including rank and they row space. The reason being that the left multiplication by M can be understood as simply being a set of (invertible) row operations and the rank and row space are properties that are preserved under (invertible) row operations. Now, as B is in row echelon form, it is easier to read this information from the matrix B, so we will use to answer the question. As B has 3 pivots, we see that the rank is 3. We can also take each row that has a pivot to form a basis of the row space, thus our basis is given by: $\left[{\begin{array}{ccccc}1&2&1&3&2\\\end{array}}\right],\left[{\begin{array}{ccccc}0&1&-3&5&-4\\\end{array}}\right],\left[{\begin{array}{ccccc}0&0&0&1&-7\\\end{array}}\right]$

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag