[Math] Let $V = \mathbb{R}^2$, show that $V$ is a vector space

linear algebravector-spaces

I am new to the concepts of vector spaces (as far as I remember), and I have some difficulties in understanding how can I show, in practice, if a set is a vector space or not.

I have an exercise in my last linear algebra assignment, which is the following:

Let $V = \mathbb{R}^2$, with addition $\oplus$ defined by:

$$\left(\begin{matrix} x \\ y \end{matrix}\right)
\oplus\left(\begin{matrix} w \\ z \end{matrix}\right) =
\left(\begin{matrix} x + w -3\\ y + z + 1 \end{matrix}\right)$$

and scalar multiplication $\odot$ defined by:

$$\alpha \odot \left(\begin{matrix} x \\ y \end{matrix}\right) =
\left(\begin{matrix} \alpha x – 3\alpha +3\\ \alpha y + \alpha – 1
\end{matrix}\right)$$

Show that $V$ is a vector space by showing that:

a) $V$ is closed with respect to $\oplus$ and $\odot$ and i.e.

  • $\forall$ v, w $\in V \rightarrow$ v $\oplus$ w $\in V$.

  • $\forall \alpha \in \mathbb{R}$, $\forall$ v $\in V \rightarrow \alpha \odot$ v $\in V$.

b) All $8$ vector spaces axioms hold. The vector $\left(\begin{matrix}
3 \\ -1 \end{matrix}\right)$ acts as a zero vector.

This can be all interesting if you have at least an idea on how to do it, and maybe you can do it, but this is not exactly my case.

I know that to show a set is a vector space I have to show it's closed under addition and under scalar multiplication (nice!). I know, from my notes, what these axioms are.

But in this case, how can I show that:

$$\left(\begin{matrix} x \\ y \end{matrix}\right) \oplus\left(\begin{matrix} w \\ z \end{matrix}\right) = \left(\begin{matrix} x + w -3\\ y + z + 1 \end{matrix}\right)$$

is closed under addition, if we don't know the values of $x$, etc?

The same for all other options (show that is closed under scalar multiplication, and show that the axioms hold).

In general, how would you prove that a set like this is a vector space? Can I see an example? I am really not understanding what I have to prove, and the examples I have tried to see for me are like Chinese.

Best Answer

Although you may not know what the values of your vector is, you do know what vectors of the set look like. In this case $\mathbb{R}^2 = \{ \begin{pmatrix} a \\ b \end{pmatrix} \colon b, a \in \mathbb{R} \}$.

Given $\begin{pmatrix} x \\ y \end{pmatrix}$ and $\begin{pmatrix} z \\ w \end{pmatrix}$, is the vector $\begin{pmatrix} x + w - 3 \\ y + z + 1 \end{pmatrix}$ in $\mathbb{R}^2$ ?

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