Space and subspace

Question

Hi I’m just staring with linear algebra, and I’d like to be ahead of schedule in my class so as to be prepared, but that also means that my question might sound a bit stupid.

I think I get what spaces and sub spaces are, at a very basic level. But I have a little problem with imagining what it does mean.

I tried to show that

$$A = \Bigg\lbrace \begin{bmatrix}x_1\\0\\0 \end{bmatrix}:\quad x_1 \in \mathbb R \Bigg\rbrace$$

Is a vector space, I showed all 8 axioms, hopefully correct ( I think it would be too long to insert here)

What I don’t really get is:

x1 is an element of $\mathbb R$, a point on a plane, but the vector-space created is a three dimensional one? So it is a three dimensional space inside $\mathbb R^3$.? Does that mean that if i can find a linear combination that goes through 0 that it is a subspace of $\mathbb R^3$? Or must a subspace be of a lower “order” like a plane or a line in this case?

Does this also imply that the zero vector is always a subspace?

Answer 1

Each vector $\begin{bmatrix}x_1\\0\\0\end{bmatrix}$ is an element of the $3$-dimensional space $\mathbb{R}^3$. However, the space that you mentioned is a $1$-dimensional one, since eqach of its elements is a multiple of $\begin{bmatrix}1\\0\\0\end{bmatrix}$. So, your space has a basis which consists of a single element.

Answer 2

The vector space described above is not three dimensional but two dimensional, even though the vectors take three coordinates. However, two of these are fixed at 0 and thus constant. The space described above is the $x_1$ axis in the three dimensional space $\mathbb{R}^3$.

Concerning your second question: a subspace needs not be of lower order than the original vector space because for every vector space $V$, it holds that $V$ is a subspace of $V$.

In your example $A$ is as mentioned before the $x_1$-axis in $\mathbb{R}^3$. Possible subspaces of $A$ are $A$ itself or $(0,0,0)$.

Answer 3

Hi many thanks for all your answers! I have different question concerning the same topic. I found different information in different books. Do Second degree polynomials form a vector space

The addition is specified as

$$ p+q=(p_0+q_0) + (p_1+q_1)x +(p_2+q_2)x^2$$

I read in some books that it isn’t a vectorspace bc the 0 vector isn’t a second degree polynomial

But than again I read that the n degrees polynomials form a vector space. I’m still trying to grasp the construct of a vector space to get a good idea of how to work with them.

Many thanks