Which of the given subsets of R3 are subspaces?
(a) The set of all vectors of the form
[a;b;0]
b)The set of all vectors of the form
[a;b;c]
where a > 0.
c) The set of all vectors of the form
[a;c;c]
d) The set of all vectors of the form
[a;b;c]
where
2a − b + c = 5.
To determine if a given subset of R3 is a subspace, we need to check three conditions:
1. The subset must contain the zero vector.
2. The subset must be closed under vector addition.
3. The subset must be closed under scalar multiplication.
Let's check each subset one by one:
(a) The set of all vectors of the form [a;b;0]:
- This subset contains the zero vector [0;0;0].
- It is closed under vector addition since adding two vectors of the form [a;b;0] will still result in a vector of the same form.
- It is closed under scalar multiplication since multiplying a vector of the form [a;b;0] by a scalar will still result in a vector of the same form.
Therefore, the subset (a) is a subspace.
(b) The set of all vectors of the form [a;b;c] where a > 0:
- This subset does not contain the zero vector since if a > 0, it cannot be equal to zero.
- It is not closed under vector addition since adding two vectors of the form [a;b;c] may result in a vector where a ≤ 0, violating the condition of the subset.
- It is not closed under scalar multiplication since multiplying a vector of the form [a;b;c] where a > 0 by a negative scalar will result in a vector where a ≤ 0, again violating the condition of the subset.
Therefore, the subset (b) is not a subspace.
(c) The set of all vectors of the form [a;c;c]:
- This subset does not contain the zero vector since if we set a = 0, the resulting vector will not be of the given form.
- It is closed under vector addition since adding two vectors of the form [a;c;c] will still result in a vector of the same form.
- It is closed under scalar multiplication since multiplying a vector of the form [a;c;c] by a scalar will still result in a vector of the same form.
Therefore, the subset (c) is a subspace.
(d) The set of all vectors of the form [a;b;c] where 2a − b + c = 5:
- This subset does not contain the zero vector since if we set a = b = c = 0, the resulting vector will not satisfy the given equation.
- It is closed under vector addition since adding two vectors of the form [a;b;c] that satisfy the given equation will still result in a vector that satisfies the equation.
- It is closed under scalar multiplication since multiplying a vector of the form [a;b;c] that satisfies the given equation by a scalar will still result in a vector that satisfies the equation.
Therefore, the subset (d) is a subspace.
In summary, the subsets (a), (c), and (d) are subspaces of R3, while the subset (b) is not.