Math 304

Test 1

Spring 2016

No books, no notes, no calculators. You must show work, unless the question is a true/false or fill-in-theblank question. Section:

Name:

Question Points Score 1

18

2

8

3

8

4

35

5

6

6

22

7

9

8

15

9

15

10

11

11

14

Total:

161

1. Fill in the blanks in the following definitions: (a) (7 points) A linear transformation F from a vector space V to a vector space W is a function from V to W that satisfies ,

(1) For all x and y in

= (2) For all x in

, and all

c =

(b) (4 points) Let V be a vector space and X = {v1 , . . . , vk } a finite set of vectors in V . The set X is called a finite basis of V if it has the following two properties: • • (c) (7 points) If V is a subset of Rn , then V is a subspace of Rn if it satisfies the following 3 conditions: 1.

∈

2. If u, v ∈

∈

, then

3. If c ∈ R and v ∈

, then

.

2. Fill in the blanks in the following statements of results from the text. (a) (4 points) If T : Rn → Rm is a linear transformation with associated matrix A, then the of A is the image T (ei ) for 1 ≤ i ≤ n. (b) (4 points) If T : Rn → Rm and S : Rm → Rs are linear transformations, with associated matrices A and B respectively, then A is a × matrix, B is a × matrix, and the matrix of the composition ST : Rn → Rs is , which is a × matrix. 3. With the matrix and vector below 

 2 1 3 A = 0 −1 −3 6 1 4 (a) (4 points) Find A−1

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  1  b = 2 3

(b) (4 points) Is b expressible as a linear combination of the columns c1 , c2 , c3 of A? Circle one: yes | no If it is, give coefficients d1 , d2 , d3 so that d1 c1 +d2 c2 +d3 c3 = b. If it is not, explain why not.

4. Indicate whether each statement is true or false by circling the appropriate answer. (a) (5 points) [ True | False ] If A is an n × n matrix with rank n, then A is invertible. (b) (5 points) [ True | False ] If a set X of vectors spans Rn , then any Y ⊂ X also has Span(Y ) = Rn . (c) (5 points) [ True | False ] If V is a subspace of Rn , then the dimension of V is at least n. (d) (5 points) [ True | False ] If (e1 , e2 , e3 ) is the standard basis of R3 and x is a vector in R4 , then there is a linear transformation T : R3 → R4 with the property that T (e1 ) = x and T (e2 ) = −x. (e) (5 points) [ True | False ] If a homogeneous system of equations in n variables has a coefficient matrix C with rank C < n, then the system has infinitely many solutions. (f) (5 points) [ True | False ] Any matrix is can be expressed as a product of elementary matrices. [N.B. The product is allowed to be of many matrices.] (g) (5 points) [ True | False ] If A is an invertible matrix, then (A−1 )T = (AT )−1 .

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5. Fill in the blanks in the following result from the text: Let A be an m × n matrix regarded as a function from Rn to Rm , and let r be the rank of A. (a) (3 points) A is onto if and only if (b) (3 points) A is one-to-one if and only if 6. Define a function T : R3 → R3 by the formula T (x1 , x2 , x3 ) = (2x2 + x3 , x1 − 4x2 , 3x1 + 6x3 ) (a) (3 points) Is T onto? (Circle one:) [ yes | no ] (b) (8 points) If your answer in the previous part was yes, explain why. If your answer in the previous part was no, find a vector in R3 which is not in the image of T .

(c) (3 points) Is T one-to-one? (Circle one:) [ yes | no ] (d) (8 points) If your answer in the previous part was yes, explain why. If your answer in the previous part was no, find two vectors in R3 whose image under T is the same.

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7. For each of the following row operations, write down an elementary matrix E such that for any 3 × 3 matrix M , EM is obtained from M by performing that row operation. (a) (3 points) Interchange the second and third rows.

(b) (3 points) Multiply the third row by 1/7.

(c) (3 points) Add four times the second row to the first row.

8. For each of the following sets of vectors in R3 , circle “yes” if the set is linearly independent, and “no” if it is not linearly independent. (a) (3 points) [ yes | no ] {(2, 1, 4), (0, 0, 0), (3, −5, 7)} (b) (3 points) [ yes | no ] {(3, 1, −2), (2, −7, 1), (12, 4, −8)} (c) (3 points) [ yes | no ] {(6, 0, −1), (2, 0, −2)} (d) (3 points) [ yes | no ] {(−2, 6, 4), (3, −9, −6)} (e) (3 points) [ yes | no ] {(1, 3, 4), (−3, 4, 7), (5, 6, −2), (7, 2, −1)} 9. For each of the following sets of vectors in R3 , circle “yes” if the set is a basis of R3 , and “no” if it is not. (a) (5 points) [ yes | no ] {(1, 2, 3), (3, 2, 1)} (b) (5 points) [ yes | no ] {(1, 2, 1), (3, 1, −5), (2, −1, 1), (−1, 0, 2)} (c) (5 points) [ yes | no ] {(1, 0, 0), (0, 2, 0), (0, 0, 5)}

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10. Let Y = {v1 , v2 } ⊂ R3 , where v1 = e1 + e2 and v2 = e2 + e3 . (a) (4 points) Show that Y is linearly independent.

(b) (4 points) Find a vector v3 which is not in the span of Y .

(c) (3 points) Let Z = {v1 , v2 , v3 }. Is Z a basis for R3 ?

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11. (14 points) Suppose B is a 2 × 2 matrix satisfying AB  4 A= 2 Find B.

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= B + I, where  2 3