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Math 232, Fall 2007
Midterm 1
Oct. 5, 2007
Last Name:
First Name:
SFU ID:
1. DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.
2. No calculators are allowed.
3. This test is comprised of 6 pages (including cover page)
4. Once the test begins, please check that all pages are intact.
5. Do ALL questions.
6. Clearly explain your answer. No credit will be given for just writing down the
answer.
7. If the answer space provided is not sufficient, write your answer on the back
of the previous page. Clearly mark the question number.
8. Good luck.
Question Points Score
Total: 50
- (7 points) Show that the vectors
are linearly independent. Show all work. SOLUTION: We form the augmented matrix and row reduce.
Since there is a pivot in every column of the coefficient matrix, we see that the vectors are linearly independent.
- (10 points) Balance the chemical equation
B 2 S 3 + H 2 O −→ H 3 BO 3 + H 2 S using the vector equation approach. Show all work. SOLUTION: We create a vector from the components Boron (B), Hydrogen (H), Sulfur (S), and Oxygen (O). We need to find positive integers x 1 , x 2 , x 3 , x 4 that give a solution to the vector equation:
x 1
+^ x^2
=^ x^3
+^ x^4
Taking the vectors on the right hand side over to the left, we get the homogeneous vector equation:
x 1
+^ x^2
+^ x^3
+^ x^4
We now form the augmented matrix and row reduce. We form the augmented matrix and row reduce. Just as in class, we see that the answer is: B 2 S 3 + 6H 2 O −→ 2H 3 BO 3 + 3H 2 S
- Let T : R^3 → R^4 be the linear transformation given by
T
x 1 x 2 x 3
x 1 + 2x 2 − x 3 3 x 2 + x 1 + 2x 3 x 3 + x 1 x 2 + x 3
(a) (3 points) Find the matrix of T. Show all work. (b) (4 points) Is T one-to-one? Justify your answer. (c) (4 points) is T onto? Justify your answer. Solution: To find the matrix of T , we compute T (~e 1 ), T (~e 2 ), and T (~e 3 ) and find that the matrix of T is
A =
To check if T is one-to-one and onto, we row reduce A.
Notice that this matrix is in echelon form; it has a pivot in every column and therefore T is one-to-one; there is not a pivot in the 4th row, and so T is not onto.
