Answer:
Check the two conditions of Subspace.
Step-by-step explanation:
If W is a Subspace of a vector space, V then it should satisft the following conditions.
1) The zero element should be in W.
Zero element can be different for different vector spaces. For examples, zero vector in is (0, 0) whereas, zero element in
is (0, 0 ,0).
2) For any two vectors, and
in W,
should also be in W.
That is, it should be closed under addition.
3) For any vector in W and for any scalar,
in V,
should be in W.
That is it should be closed in scalar multiplication.
The conditions are mathematically represented as follows:
1) 0 W.
2) If then
.
3)
Here V = and W = Set of all (x, y, z) such that
We check for the conditions one by one.
1) The zero vector belongs to the subspace, W. Because (0, 0, 0) satisfies the given equation.
i.e., 0 - 2(0) + 5(0) = 0
2) Let us assume and
are in W.
That means: and
We should check if the vectors are closed under addition.
Adding the two vectors we get:
Rearranging these terms we get:
So, the equation becomes, 0 + 0 = 0
So, it s closed under addition.
3) Let k be any scalar in V. And
This means
Taking k common outside, we get:
The equation becomes k(0) = 0.
So, it is closed under scalar multiplication.
Hence, W is a subspace of .