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)
1
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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