Problem Prove that the real vector space of all continous real-valued functions on the interval $[0,1]$ is infinite-dimensional Solution Construct the linearly independent sequence of functions in $\mathbb{R}^{[0,1]}$ $1,z,z^2,z^3, \dots$ where the $j$‘th vector in the sequence is the function $z^{j-1}$. The existence of such a sequence then implies, by P2A14, that $\mathbb{R}^{[0,1]}$ is infinite dimensional.

Problem Prove that $\mathbb{F}^\infty$ is infinite dimensional Solution Construct the infinite sequence of linearly independent vectors in $\mathbb{F}^\infty$ $(1,0,0,\dots), (0,1,0,\dots), \dots$ where the $j$‘th vector in the sequence has a one in the $j$‘th coordinate and zeros everywhere else. The existence of such a sequence then implies, by P2A14, that $\mathbb{F}^\infty$ is infinite dimensional.

Problem Prove that $V$ is infinite-dimensional if and only if there is a sequence $v_1, v_2, \dots$ of vectors in $V$ such that $v_1, \dots, v_m$ is linearly independent for every positive integer $m$ Solution $(\rightarrow)$ $V$ being infinite-dimensional implies that we can always find a $v_{m+1}$ that is not in the span of linearly independent vectors $v_1, \dots, v_m$ for all positive integers $m$. By P2A11 we can therefore construct a sequence $v_1, v_2, \dots$ of vectors with the property that for any $m \in \mathbb{N}$ we have linear independence for $v_1, v_2, \dots, v_m$....

Problem Explain why no list of four polynomials spans $\mathcal{P}_4(\mathbb{F})$ Solution The list $1,z,z^2,z^3,z^4$ has five linearly independent elements and spans $\mathcal{P}_4(\mathbb{F})$, therefore by theorem $2.23$ no list of less than $5$ polynomials can span the space.

Problem Explain why there does not exist a list of six polynomials that is linearly independent in $\mathcal{P}_4(\mathbb{F})$ Solution The list $1,z,z^2,z^3,z^4$ has five linearly independent elements and spans $\mathcal{P}_4(\mathbb{F})$, therefore by theorem $2.23$ no list with more than 5 elements that spans the space can be linearly independent.