Anjal Thakuri MAT210Fa20Yuan Assignment HomeworkAssignment9 due 11/11/2020 at 11:59pm EST

1. (1 point) Determine if the subset of R3 consisting of vectors of the

form

ab c

, where a≥ 0, b≥ 0, and c≥ 0 is a subspace. Select true or false for each statement. ? 1. This set is a subspace ? 2. This set is closed under vector addition ? 3. This set is closed under scalar multiplications ? 4. The set contains the zero vector Answer(s) submitted: • • • •

(incorrect)

2. (1 point) Determine if the subset of R3 consisting of vectors of the

form

ab c

, where at most one of a, b, and c is nonzero, is a subspace.

Select true or false for each statement. ? 1. This set is closed under scalar multiplications ? 2. The set contains the zero vector ? 3. This set is closed under vector addition ? 4. This set is a subspace Answer(s) submitted: • • • •

(incorrect)

3. (1 point) Determine if the subset of R2 consisting of vectors of the

form

v1… vn

, where v1− v2 + v3− v4 + v5−·· ·− vn = 0 is a subspace. Select true or false for each statement. ? 1. This set is a subspace ? 2. This set is closed under scalar multiplications ? 3. This set is closed under vector addition ? 4. The set contains the zero vector Answer(s) submitted:

• • • •

(incorrect)

4. (1 point) Determine if the subset of R2 consisting of vectors of the

form [

a b

] , where a+b = 1 is a subspace.

Select true or false for each statement.

? 1. This set is closed under scalar multiplications ? 2. This set is a subspace ? 3. The set contains the zero vector ? 4. This set is closed under vector addition Answer(s) submitted:

• • • •

(incorrect)

5. (1 point) Indicate whether the statement is true or false.

? 1. If T : R4 → R8 is a linear transformation, then range (T ) is a subspace of R8.

Answer(s) submitted:

• (incorrect)

6. (1 point) Determine if the subset of R4 consisting of vectors of the

form

a

3a+b −4a−5b −5a−5b

is a subspace. Select true or false for each statement. ? 1. This set is closed under vector addition ? 2. This set is closed under scalar multiplications ? 3. The set contains the zero vector ? 4. This set is a subspace Answer(s) submitted:

• • • •

(incorrect)

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7. (1 point) Determine if the subset of R2 consisting of vectors of the

form [

a b

] , where a and b are integers, is a subspace.

Select true or false for each statement.

? 1. This set is closed under vector addition ? 2. This set is a subspace ? 3. This set is closed under scalar multiplications ? 4. The set contains the zero vector

Answer(s) submitted:

• • • •

(incorrect)

10. (1 point) Which of the following sets are subspaces of R3?

• A. {(x,y,z) | −2x+5y−7z =−3} • B. {(3x,2x,−5x) | x arbitrary number } • C. {(x,y,z) | x+ y+ z = 0} • D. {(6,y,z) | y,z arbitrary numbers } • E. {(x,y,z) | −6x+4y = 0,−9x+3z = 0} • F. {(x,y,z) | x < y < z}

Answer(s) submitted:

•

(incorrect)

11. (1 point)

Let V = R2 and let H be the subset of V of all points on the line 3x−2y = −6. Is H a subspace of the vector space V ?

(1) Is H nonempty?

• choose • H is empty • H is nonempty

(2) Is H closed under addition? If it is, enter CLOSED . If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as <1,2>, <3,4>.

(3) Is H closed under scalar multiplication? If it is, enter CLOSED . If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, <3,4>.

(4) Is H a subspace of the vector space V ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.

• choose • H is a subspace of V • H is not a subspace of V

Answer(s) submitted:

• • • •

(incorrect)

12. (1 point) 2

Let V = R2 and let H be the subset of V of all points on the line 3x− 4y = 0. Is H a subspace of the vector space V ?

(1) Does H contain the zero vector of V ?

• choose • H contains the zero vector of V • H does not contain the zero vector of V

(2) Is H closed under addition? If it is, enter CLOSED . If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as <1,2>, <3,4>.

(3) Is H closed under scalar multiplication? If it is, enter CLOSED . If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, <3,4>.

(4) Is H a subspace of the vector space V ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.

• choose • H is a subspace of V • H is not a subspace of V

Answer(s) submitted:

• • • •

(incorrect)

13. (1 point)

Let V = R2 and let H be the subset of V of all points in the first and third quadrants that lie between the lines y = 2x and y = x/2. Is H a subspace of the vector space V ?

(1) Does H contain the zero vector of V ?

• choose • H contains the zero vector of V • H does not contain the zero vector of V

(2) Is H closed under addition? If it is, enter CLOSED . If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as <1,2>, <3,4>.

(3) Is H closed under scalar multiplication? If it is, enter CLOSED . If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, <3,4>.

(4) Is H a subspace of the vector space V ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.

• choose • H is a subspace of V • H is not a subspace of V

Answer(s) submitted: • • • •

(incorrect)

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