Is span of the vectors $\{v_1, v_2, 0\}$ equivalent to the span of $\{v_1, v_2\}$ ?
Im struggling to think whether this statement is true or not mainly because my train of thought is:
all vectors in a span can be multiplied by $0$ to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?
On
The span of $\{v_1,v_2,0\}$ is the set of all linear combinations of these three vectors, and this certainly equals the set of all linear combinations of $v_1,v_2$ which is the span of $\{v_1,v_2\}$.
Perhaps you have confused this with another matter in your question: Indeed, the zero vector is in the span of any set of vectors, because by using zero coefficients for each vector we generate the zero vector. You need to distinguish between (1) the span of a set of vectors and (2) the vectors in the set used to span.
On
Yes!
Span operation is monotone!
That is if $A \subset B$ then $\text{span}(A) \subset \text{span}(B)$
So $\text{span}(\{v_1,v_2\}) \subset \text{span}(\{v_1,v_2,0\})$
The reverse inclusion is also true, since.....?
Remembert that $\text{span}(A)$ means intersection of all subspaces containing $A$ and since this intersection is again a subspace, zero is in that space . So we dont worrying about including zero in span
On
all vectors in a span can be multiplied by 0 to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?
It's not entirely clear what your thinking is. What you might be getting at is that given any $v_1,v_2...v_n$, if $v_{n+1}$ is in the span of $\{v_1,v_2...v_n\}$, then the span of $\{v_1,v_2...v_n\}$ and the span of $\{v_1,v_2...v_n,v_{n+1}\}$ are the same. The span of a set of vectors is, by definition, closed over linear combinations; adding $v_{n+1}$ to your list of vectors doesn't add anything new to the span, if you can get $v_{n+1}$ from the other vectors.
In the example you give, the span of $\{v_1,v_2,0\}$ is defined as the set of all vectors that can be written in the form $c_1v_1+c_2v_2+c_30$, for some scalars $c_1,c_2,c_3$. But if $v=c_1v_1+c_2v_2+c_30$, then clearly $v=c_1v_1+c_2v_2$, so $v$ is in the span of $\{v_1,v_2\}$
Yes of course by definition it easy to check that the sets $\{v_1, v_2, 0\}$ and $\{v_1, v_2\}$ span the same subspace and that the zero vector is always in the span of any non empty set of vectors.