Are the following vector spaces with the indicated operations?
(a)
The set \(V\) of all polynomials of degree greater than or equal to 3, together with 0; operations of \(\mathbb {P}\).
\(V\) is a vector space. \(V\) is not a vector space.
(b)
The set \(V\) of all polynomials of degree less than or equal to 3; operations of \(\mathbb {P}\).
\(V\) is a vector space. \(V\) is not a vector space.
(c)
The set \(V\) of \(2 \times 2\) matrices with zero determinant; usual matrix operations.
\(V\) is a vector space. \(V\) is not a vector space.
(d)
The set \(V\) of all \(2 \times 2\) matrices whose entries sum to 0; usual matrix operations.
\(V\) is a vector space. \(V\) is not a vector space.

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 6.1.2.