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.
please help!
To determine whether a given subset of R3 is a subspace, we need to check three conditions: closure under addition, closure under scalar multiplication, and the presence of the zero vector.
(a) The set of all vectors of the form [a;b;0]:
To check closure under addition, take two vectors, say [a1; b1; 0] and [a2; b2; 0], and add them together. The result is [a1 + a2; b1 + b2; 0]. Since the third component is always 0, closure under addition is satisfied.
To check closure under scalar multiplication, take any vector [a; b; 0] and multiply it by a scalar k. The result is [ka; kb; 0]. Again, the third component remains 0, so closure under scalar multiplication is satisfied.
The zero vector, [0; 0; 0], is present in this subset.
Therefore, the subset of all vectors of the form [a; b; 0] is a subspace of R3.
(b) The set of all vectors of the form [a; b; c] where a > 0:
To check closure under addition, take two vectors [a1; b1; c1] and [a2; b2; c2]. Adding them gives [a1 + a2; b1 + b2; c1 + c2]. Since the a-component does not necessarily remain positive, closure under addition is not satisfied.
Therefore, this subset is not a subspace of R3.
(c) The set of all vectors of the form [a; c; c]:
Similar to part (b), closure under addition is not satisfied as adding two vectors would result in [a1 + a2; c1 + c2; c1 + c2]. Since the a-component is not restricted, this subset does not fulfill closure under addition.
Therefore, this subset is not a subspace of R3.
(d) The set of all vectors of the form [a; b; c] where 2a − b + c = 5:
To check closure under addition, take two vectors [a1; b1; c1] and [a2; b2; c2]. Adding them gives [a1 + a2; b1 + b2; c1 + c2]. Since the sum of the a-components is not necessarily equal to 5, closure under addition is not satisfied.
Therefore, this subset is not a subspace of R3.
In summary, out of the given subsets:
(a) The set of all vectors of the form [a; b; 0] is a subspace of R3.
(b), (c), and (d) are not subspaces of R3.