Vector Spaces

[Faded-Maximus]

Let H be a subset of R3, defined by: W={(x, y, z) : x - y + 1 = e^z}. Is H a subspace of R3?

To determine if H is a subspace of R3, we need to satisfy the three properties.

The zero vector of R3 is in H because substituting x = y = z = 0 into x - y + 1 = e^0 gives us a valid expression.

I am not sure how to determine whether H is closed under vector addition or multiplication though.

 

[pka]

Think about scalar multiplication.
Can you show that the set is closed with respect to scalar multiplication.

 

[Faded-Maximus]

I'm not sure whether or not this is correct, but I am going to go with no.

A counter-example could be.
Let x = y = z = 1
Let c = 5

which would give us cx - cy + 1 = e^cz
5 - 5 + 1 = e^(5)(1)
1 = e^5 which is not true.