What are the three properties you need to verify? You didn't list them in the 2nd section or anywhere else.yango_17 said:Homework Statement
Is the set ##W## a subspace of ##\mathbb{R}^{3}##?
##W=\left \{ \begin{bmatrix}
x\\
y\\
z
\end{bmatrix}:x\leq y\leq z \right \}##
Homework Equations
The Attempt at a Solution
I believe the set is indeed a subspace of ##\mathbb{R}^{3}##, since it looks like it will satisfy the three properties of a subspace. I'm wondering as to how one would go about explicitly proving this. Any help would be appreciated. Thanks.
Let, say, ##\vec{u}## and ##\vec{v}## be elements of W, with ##\vec{u} = <u_1, u_2, u_3>## and ##\vec{v} = <v_1, v_2, v_3>##.yango_17 said:So, the properties would be that the set needs to contain the zero vector, needs to be closed under scalar multiplication, and needs to be closed under addition. When you say take arbitrary members of the set to test the last two properties, what exactly do you mean?
A subspace of ##\mathbb{R}^{3}## is a subset of ##\mathbb{R}^{3}## that satisfies three conditions: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication. In other words, a subspace is a smaller vector space that is contained within a larger vector space.
To check if a set is a subspace of ##\mathbb{R}^{3}##, you need to verify that it satisfies the three conditions mentioned above. This means checking if the set contains the zero vector, if it is closed under vector addition, and if it is closed under scalar multiplication. If all three conditions are met, then the set is a subspace of ##\mathbb{R}^{3}##.
Yes, a subspace of ##\mathbb{R}^{3}## can be a line or a plane. As long as the set satisfies the three conditions of a subspace, it can be any size or shape within ##\mathbb{R}^{3}##. For example, the x-y plane or the line y = 2x are both subspaces of ##\mathbb{R}^{3}##.
A subspace is a subset of a vector space that satisfies certain conditions, while a span is the set of all possible linear combinations of a given set of vectors. In other words, a subspace is a specific type of set within a vector space, while a span is a mathematical concept used to describe a set of vectors.
No, a set cannot be a subspace of ##\mathbb{R}^{3}## if it does not contain the zero vector. The zero vector is a necessary condition for a set to be a subspace, as it is required for the set to be closed under scalar multiplication and vector addition. Without the zero vector, the set cannot be considered a subspace of ##\mathbb{R}^{3}##.