PROBLEM:Determine if the subset is a subspace of the indicated vector space. Give reasons for your answers.

1. The zero vector together with all vectors $\displaystyle x=(x_1,x_2)$ in $\displaystyle \mathbb{R}^2$ for which $\displaystyle \frac{x_2}{x_1}$ has a constant value.

ATTEMPT:

The book states that the subset satisfies the subspace criterion, therefore it is a subspace of the indicated vector space. However,I have reason to believe it to be otherwise. First,

$\displaystyle W=(x| 0+x);\frac{x_2}{x_1}$ and $\displaystyle x=(x_1, x_2)$.

Here, in set notation, I have defined that the subset in question, W, has a zero vector "together" with all vectors, x. By "together", I presume they mean the linear combination of the zero vector with all vectors.

Next, I observe that $\displaystyle x_2/x_1$ has a constant value.What does it mean to be a constant value?I asked myself. To be a constant value, $\displaystyle x_2$ and $\displaystyle x_1$ must also be constant values; they do not contain variable terms. In addition, if we suppose that $\displaystyle x_1$ be equal to 0, then the quotient is undefined—which also is not a constant value. The latter is my argument for whythis subset is not a subspace of the indiciated vector space.

Example:

The vector x = 0 + (0,2) would not belong in W because $\displaystyle \frac{1}{0}$ is undefined. Which also qualifies that it is not a constant value.

If we imagined two vectors which do belong in W, namely x = 0 + (1,1) and y = 0 + (-1,1). These two vectors indeed belong in W because (1,1) and (-1,1) are vectors in $\displaystyle \mathbb{R}^2$ that have constant values $\displaystyle x_2/x_1$ together with the zero vector.

However, the linear combination of the two produces a vector that does not belong in W, thereby violating the subspace critera,

Subspace criterion.If every vector of the form

$\displaystyle {\alpha}_1x_1+{\alpha}_2x_2$belongs to W whenever $\displaystyle x_1$ and $\displaystyle x_2$ belong to W, and $\displaystyle {\alpha}_1$ and $\displaystyle {\alpha}_2$ are arbitrary sclars, then W is a subspace of V.

I take this rule strictly because the subspace criterion states that is is a subspace ifeveryvector is of the form.

Thoughts?