Question

is this set a vector space or not?

Answer 1

\[\left(\begin{matrix}1 \\ 0\end{matrix}\right),\left(\begin{matrix}2 \\ 0\end{matrix}\right),\left(\begin{matrix}3 \\ 0\end{matrix}\right)\]

Answer 2

if it is, then, where is identity vector ?

Answer 3

I really need someone to explain. I am not getting what a vector space means

Answer 4

read your book, definition of vector space part

Answer 5

need my book?

Answer 6

literally says, sum of any two element of set S is in S and product of arbitrary scalar and element is in S.
But i really need explanation, this is not enough

Answer 7

I vector space is a set of vectors along with two operations (addition and scalar multiplication) such that a certain number of properties are satisfied. One of those properties, and @OOOPS mentioned, is that it must contain the "zero vector" does this set include such a vector?

Answer 8

did it say subspace or vector space?

Answer 9

vector space

Answer 10

either way the zero vector MUST be there.

Answer 11

no there is no zero vector but why do we need a zero vector

Answer 12

but there is additive inverse of these vector. correct?

Answer 13

read axiom A, 3, you still have part C, but it is enough to say your set is not a vector space

Answer 14

Ahh I see, so if we add vector \[\left(\begin{matrix}0 \\ 0\end{matrix}\right)\] will this be a vector space

Answer 15

no

\(\left(\begin{matrix}2 \\ 0\end{matrix}\right)+\left(\begin{matrix}3 \\ 1\end{matrix}\right)=\left(\begin{matrix}5 \\ 0\end{matrix}\right) \notin\{\left(\begin{matrix}0 \\ 0\end{matrix}\right),\left(\begin{matrix}2 \\ 0\end{matrix}\right),\left(\begin{matrix}3 \\ 0\end{matrix}\right)\} \)

Answer 16

so it is also not closed with respect to vector addition.

Answer 17

@best_mathematician do you understand?

Answer 18

yes just made much more sense. Thank you

Answer 19

of course that should say

\(\left(\begin{matrix}2 \\ 0\end{matrix}\right)+\left(\begin{matrix}3 \\ 0\end{matrix}\right)=\left(\begin{matrix}5 \\ 0\end{matrix}\right) \notin\{\left(\begin{matrix}0 \\ 0\end{matrix}\right),\left(\begin{matrix}2 \\ 0\end{matrix}\right),\left(\begin{matrix}3 \\ 0\end{matrix}\right)\}\)

Answer 20

np