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Which one of the following is a subspace of the vector space R3 ?
- a){(x, y, z) ∈ R3 : x + 2y = 0, 2x + 3z = 0}
- b){(x, y, z) ∈ R3 : 2x + 3y + 4z – 3 = 0, z = 0}
- c){(x, y, z) ∈ R3: x ≥ 0, y ≥ 0}
- d){(x, y, z) ∈ R3 : x – 1 = 0, y = 0}
Correct answer is option 'A'. Can you explain this answer?
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Which one of the following is a subspace of the vector space R3?a){(x,...
Subspaces in R3
In order to determine which of the given options is a subspace of the vector space R3, we need to check if each option satisfies the three conditions required for a subspace:
1. The zero vector is in the subspace.
2. The subspace is closed under vector addition.
3. The subspace is closed under scalar multiplication.
Option A: {(x, y, z) R3 : x 2y = 0, 2x 3z = 0}
Let's check if this option satisfies the three conditions:
1. Zero vector: To check if the zero vector (0, 0, 0) is in the subspace, we substitute x = 0, y = 0, and z = 0 into the given equations:
- 0 + 2(0) = 0, which is true.
- 2(0) + 3(0) = 0, which is also true.
Therefore, the zero vector is in the subspace.
2. Vector addition: Let (x1, y1, z1) and (x2, y2, z2) be two vectors in the subspace. We need to show that their sum, (x1 + x2, y1 + y2, z1 + z2), is also in the subspace.
We substitute the values into the given equations:
- (x1 + x2) + 2(y1 + y2) = (x1 + 2y1) + (x2 + 2y2) = 0 + 0 = 0, which is true.
- 2(x1 + x2) + 3(z1 + z2) = 2x1 + 2x2 + 3z1 + 3z2 = 2(0) + 2(0) = 0, which is also true.
Therefore, the subspace is closed under vector addition.
3. Scalar multiplication: Let (x, y, z) be a vector in the subspace and c be a scalar. We need to show that the scalar multiple c(x, y, z) is also in the subspace.
We substitute the values into the given equations:
- c(x) + 2c(y) = cx + 2cy = 0, which is true.
- 2c(x) + 3c(z) = 2cx + 3cz = 0, which is also true.
Therefore, the subspace is closed under scalar multiplication.
Since option A satisfies all three conditions, it is a subspace of the vector space R3.
In order to determine which of the given options is a subspace of the vector space R3, we need to check if each option satisfies the three conditions required for a subspace:
1. The zero vector is in the subspace.
2. The subspace is closed under vector addition.
3. The subspace is closed under scalar multiplication.
Option A: {(x, y, z) R3 : x 2y = 0, 2x 3z = 0}
Let's check if this option satisfies the three conditions:
1. Zero vector: To check if the zero vector (0, 0, 0) is in the subspace, we substitute x = 0, y = 0, and z = 0 into the given equations:
- 0 + 2(0) = 0, which is true.
- 2(0) + 3(0) = 0, which is also true.
Therefore, the zero vector is in the subspace.
2. Vector addition: Let (x1, y1, z1) and (x2, y2, z2) be two vectors in the subspace. We need to show that their sum, (x1 + x2, y1 + y2, z1 + z2), is also in the subspace.
We substitute the values into the given equations:
- (x1 + x2) + 2(y1 + y2) = (x1 + 2y1) + (x2 + 2y2) = 0 + 0 = 0, which is true.
- 2(x1 + x2) + 3(z1 + z2) = 2x1 + 2x2 + 3z1 + 3z2 = 2(0) + 2(0) = 0, which is also true.
Therefore, the subspace is closed under vector addition.
3. Scalar multiplication: Let (x, y, z) be a vector in the subspace and c be a scalar. We need to show that the scalar multiple c(x, y, z) is also in the subspace.
We substitute the values into the given equations:
- c(x) + 2c(y) = cx + 2cy = 0, which is true.
- 2c(x) + 3c(z) = 2cx + 3cz = 0, which is also true.
Therefore, the subspace is closed under scalar multiplication.
Since option A satisfies all three conditions, it is a subspace of the vector space R3.
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Which one of the following is a subspace of the vector space R3?a){(x,...
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Which one of the following is a subspace of the vector space R3?a){(x, y, z) ∈R3 : x + 2y = 0, 2x + 3z = 0}b){(x, y, z) ∈R3 : 2x + 3y + 4z – 3 = 0, z = 0}c){(x, y, z)∈ R3: x ≥0, y ≥ 0}d){(x, y, z) ∈ R3 : x – 1 = 0, y = 0}Correct answer is option 'A'. Can you explain this answer?
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