MT150 Mathematical Foundations Semester 2, 2000
 
 

• These questions are taken from Strang, "An introduction to Linear Algebra".
• The sections listed contain further questions relevant to much of the material in MT150
• Answers to many of the questions are given in the book.

1. Draw the vectors v =  and w =  and v + w and v - w in a single xy plane.

2. If v + w =  and v - w =  compute and draw v and w.

3. From v =  and w =  find the components of 3v + w , v - 3w , and cv + dw.

GS Section 1.2

1. Calculate the dot products 

u =  v =  w = .

2. Compute the lengths  of those vectors. Check the Schwarz inequalities

3. Write down the unit vectors in the directions of v and w in problem 1 . Find the cosine of the angle  between them.

GS Section 2.1

9. Compute each Ax by dot products of the rows with each column vector.

(a)  (b) 

GS Section 2.2

5. For the second equation below choose a right side which gives a system with no solution and another right side which gives a system with infinitely many solutions. What are two of those solutions?
 
 

GS Section 2.4

1. Suppose A is 3 by 5, B is 5 by 3, C is 5 by 1 and D is 3 by 1. Which of these matrix operations are allowed and what are the shapes of the results?

(a) BA ; (b) A(B+C) ; (c) ABD ; (d) AC + BD ; (e) ABABD.

2. What rows or columns and what matrices do you multiply to find

(a) the third column of AB? (b) the first row of AB? (c) the entry in row3 , column 4 of AB? (d) the entry in row1, column 1 of CDE.

6. Show that (A+B)² is different from A² + 2AB + B² when
 
 

Write down the correct rule for (A+B)(A+B) = A² + AB + _____ + B².

Additional exercises

A1. Let a =  , b = , c =  and d = . Find x, y, and z such that xa + yb + zc = d.

A2. Solve the following systems by systematic elimination

(a)      x + 3y + z = 7                     (b)      2x + 8y + 3z = -7

       3x + 8y +2z = 18                                      4y + 2z = b

         2x + 4y + z = 9                                   x + 3y + z = -2

In part (b) consider the cases b = - 6 and b not equal to - 6.

A3. (a) Evaluate the following determinants 

(b) Find any values of c for which the determinant below is zero
 
 

(c) Let  .

Verify that d(1,y) = 0 , that d(x,1) = 0 , and that d(x,x) = 0. Hence or otherwise express d(x,y) as a product of 3 factors linear in x or y .

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