Example SVP4 leaves several details for the reader to check. Verify these five claims.

Determine if the set $T=\set{x^2-x+5,\,4x^3-x^2+5x,\,3x+2}$ spans the vector space of polynomials with degree 4 or less, $P_4$. (Compare the solution to this exercise with Solution LISS.C40.)

Trivially, if $U$ and $V$ are two subspaces of $W$ with $U = V$, then $\dimension{U}=\dimension{V}$. Combine this fact, Theorem PSSD, and Theorem EDYES all into one grand combined theorem. You might look to Theorem PIP for stylistic inspiration. (Notice this problem does not ask you to prove anything. It just asks you to roll up three theorems into one compact, logically equivalent statement.)

Prove the following theorem, which could be viewed as a reformulation of parts (3) and (4) of Theorem G, or more appropriately as a corollary of Theorem G (Proof Technique LC).

Suppose $V$ is a vector space and $S$ is a subset of $V$ such that the number of vectors in $S$ equals the dimension of $V$. Then $S$ is linearly independent if and only if $S$ spans $V$.

Suppose that $A$ is an $m\times n$ matrix and let $\text{min}(m,\,n)$ denote the minimum of $m$ and $n$. Prove that $\rank{A}\leq \text{min}(m,\,n)$. (If $m$ and $n$ are two numbers, then $\text{min}(m,\,n)$ stands for the number that is the smaller of the two. For example $\text{min}(4,\,6)=4$.)

Suppose that $A$ is an $m\times n$ matrix and $\vect{b}\in\complex{m}$. Prove that the linear system $\linearsystem{A}{\vect{b}}$ is consistent if and only if $\rank{A}=\rank{\augmented{A}{\vect{b}}}$.

Suppose that $V$ is a vector space with finite dimension. Let $W$ be any subspace of $V$. Prove that $W$ has finite dimension.

Part of Exercise B.T50 is the half of the proof where we assume the matrix $A$ is nonsingular and prove that a set is a basis. In Solution B.T50 we proved directly that the set was both linearly independent and a spanning set. Shorten this part of the proof by applying Theorem G. Be careful, there is one subtlety.

Suppose that $W$ is a vector space with dimension 5, and $U$ and $V$ are subspaces of $W$, each of dimension 3. Prove that $U\cap V$ contains a nonzero vector. State a more general result.