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Did you read my tips on how to study and learn Math 1310? If not, here is a link to those important suggestions: Did you see my tips for Assignment 1? Did you see my tips for Assignment 2? Did you see my tips for Assignment 3? Did you see my tips for Assignment 4? Here is a link to the actual assignment, in case you don't have it handy: You need to study Lesson 14 (Vectors), Lesson 15
(Equations of Lines in Euclidean 3-Space), Lesson 16 (Vector Spaces and Subspaces), Lesson 17 (Linear Independence), and Lesson 18 (Basis and Dimension) from my Matrices for Management book to prepare for this assignment.
PLEASE NOTE: I have totally re-jigged my Matrices for Management book this year. Two new lessons have been added, and I have changed the order of many of the other lessons to better fit the syllabus for the distance course. I always appreciate it when students purchase the current edition of my books, as that does subsidize my efforts to offer things like these free tips. If you choose to make do with an older edition of my book, please realize
that you are missing some things, but you can probably make do. I leave it up to you to look at the Table of Contents in my free sample of the book above to clarify what lesson number in your old book pertains to the lessons I mention in my tips.
Don't have my book? You can
download a free sample of my book and audio lectures containing Lessons 1, 2 and 3: Here is a link where you can download my lesson on Equations of Lines if you are using an older edition of my book to assist you with question 1 in the assignment. Questions 1, 2 and 3 in my lesson are similar. By the way, if you are using
an older edition of my book, why not consider purchasing my audio lectures to help with your studies, and to show some appreciation for my assistance through this course? Mention the tips in your email request, and I will knock $10 off the price (meaning you will pay only $30 + GST = $31.50.) This is not unlike my examples in Lesson 14.
Question 2(b) is just a matter of isolating s in the equation algebraically first.
Note that, when they put vertical lines around an expression (like absolute value
signs), as in 2(c), that is what I would denote with double lines. In other words, |u-v| is what I would denote ||u-v||, which means they want the length or norm of u-v.
I define the nullspace in my Lesson 18, questions 3 and 4.
The easiest way to do this question is to transpose the given vector in each part and call that transpose vector x. Leave the original matrix, A, alone, transpose the vectors
u1, etc. Which is to say, write those vectors down one column rather than the row form they are given as. Then check if Ax = 0, where 0 is a zero vector. If and only if, the answer is "yes", the given vector must be in the nullspace. For example, if u1 is in the nullspace, then the product of A and u1-transpose, will be a zero vector.
Note
that, if it is impossible to multiply A by the transpose of the vector, that certainly proves it is not in the nullspace.
This is more like my Lesson 18, questions 3 and 4.
Very similar to my questions in Lesson 16. Feel free to do this question open-book using similar examples from my book. Very unlikely you will ever have to do this kind of thing on your exam. Their questions are quite similar to my
question 1(a) and (b).
This is back to Lesson 14 again. You might find my question 2 helpful. Be careful though!
Make sure you have studied the Linear Independence section in Lesson 17 of my book and gone through my question 4 in that lesson. Hint: you can use a determinant to solve this problem. Note that it is also quite similar to my
Lesson 10, question 5. What must the determinant equal if the vectors are linearly dependent?
Very similar to my Lesson 17, questions 1, 2 and, especially, 3. Your answers for part (a) and (b) will tell you the answer to part (c). Make sure you have learned the definition of a basis from Lesson 18 in my book.
Questions 5 and 6 in Lesson 18 give you some specific examples that might help with part (c) as well.
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