Usual operations in a vector space

Question

The set $$W=\left\{\left(x,y,z\right) \in \mathbb{R}^3 | x+y+z=0 \text{ and } 2x+y=0\right\}$$ with the usual operations is a vector space?

What is "usual operations"?

Answer 1

The usual operations they are referring to is the "addition" of vectors, typically notated with a $+$ is defined as

$$(A,B,C)+(a,b,c) = (A+a,B+b,C+c)$$

where the $+$ in $(A,B,C)+(a,b,c)$ is the newly defined addition of vectors while the $+$'s inside of $(A+a,B+b,C+c)$ is the addition that you are already familiar with for adding real numbers together.

The other operation here is the "multiplication" of a vector by a scalar, typically notated with a $\cdot$ is defined as

$$\alpha\cdot (a,b,c) = (\alpha \cdot a, \alpha \cdot b,\alpha \cdot c)$$

where the $\cdot$ in $\alpha\cdot (a,b,c)$ is the newly defined scalar multiplication that multiplies a scalar with a vector, and the $\cdot$'s inside of $(\alpha\cdot a,\alpha\cdot b,\alpha \cdot c)$ is the multiplication of real numbers that you are already familiar with.

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Note that here they emphasize that these are what the addition and multiplication are for this problem because this doesn't need to always be the case. You can have more exotic operations in use than these.