Basis and dimension

Let $\mathbf{P}_{3}$ be the vector space of polynomials of degree at most 3. Determine which of the following are bases for this vector space.

1. $\left\{x^{3}+1,~{}x^{2}+x,~{}2x^{3}+x^{2},~{}2x^{3}-x^{2}-3x+1\right\}$.

2. $\left\{x+1,~{}x^{3}+x^{2}+2x,~{}x^{2}+x,~{}x^{3}+x^{2}+x\right\}$.

Let $V$ be a $5$-dimensional vector space. If you have $5$ linearly independent vectors in $V$, can you conclude that the vectors span $V$?

Let $V$ be a $5$-dimensional vector space. If you have $6$ vectors in $V$, is it possible that they are linearly independent? Explain.

1. Let $k$ be a positive integer, let $W_{k}\subseteq\mathbf{Seq}_{K}$ be the subspace consisting of all sequences that are periodic with period $k$ (see Exercise 38). Find a basis for $W_{k}$. What is its dimension?

2. More difficult: find a basis for the infinite-dimensional vector space consisting of all periodic sequences of all periods.

Let $K=\mathbb{Q}$, the field of rational numbers. Consider vectors of the form $a+b\sqrt{2}$ where $a,b$ are rational numbers. Show that this collection of vectors is a vector space over $\mathbb{Q}$ and give a basis for this vector space. What is its dimension?