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 in for which has a constant value.
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,
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 has a constant value. What does it mean to be a constant value? I asked myself. To be a constant value, and must also be constant values; they do not contain variable terms. In addition, if we suppose that be equal to 0, then the quotient is undefined—which also is not a constant value. The latter is my argument for why this subset is not a subspace of the indiciated vector space.
The vector x = 0 + (0,2) would not belong in W because 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 that have constant values 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
belongs to W whenever and belong to W, and and 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 if every vector is of the form.