Subspace of strictly positive entries Can someone show me how to solve this question.


  Is the set of all vectors in $R^3$ with strictly positive entry a subspace?


Thanks
To determine whether something is a linear a subspace you have to verify that it is itself a vector space.

There are usually about two things that go wrong: Either $0$ is not in the space or $x,y$ are in the space but $x+y$ is not.

This later "flaw" is generally referred to as "not being closed" which is short for "not closed with respect to vector addition".

These are the two things that go wrong most often but sometimes it also happens that there are no inverses. After, first and foremost, a vector space is a group and if it does not satisfy the group axioms it cannot be a group. 

In your example, two of the three things I mention go wrong. Do you see which?
Is the set of all vectors in $\mathbb{R}^3$ with strictly positive entry a subspace?


Let
$$
\mathbb{R}_{&gt;0}^3 := \left\{ \left. \left( x,y,z \right)^\mathsf{T} \in \mathbb{R}^3 \,\right|\, x &gt; 0 \wedge y &gt; 0 \wedge z &gt; 0 \right\}
$$
denote the set of all vectors with strictly positive entries. It is a subset of $\mathbb{R}^3$.

In order for a subset to be a subspace of a given vector space, it is required that it contains the zero vector of the original vector space.

The zero vector $\left( 0,0,0 \right)^\mathsf{T} \in \mathbb{R}^3$ is, however, not in $\mathbb{R}_{&gt;0}^3$. Therefore, $\mathbb{R}_{&gt;0}^3$ is not a subspace of $\mathbb{R}^3$.