Define a possibly new vector space by beginning with the set and vector addition from $\complex{2}$ (Example VSCV) but change the definition of scalar multiplication to
\begin{align*}
\alpha\vect{x}&=\zerovector=\colvector{0\\0}&&\alpha\in\complex{\null},\ \vect{x}\in\complex{2}
\end{align*}
Prove that the first nine properties required for a vector space hold, but Property O does not hold.
This example shows us that we cannot expect to be able to derive Property O as a consequence of assuming the first nine properties. In other words, we cannot slim down our list of properties by jettisoning the last one, and still have the same collection of objects qualify as vector spaces.
Let $V$ be the set $\complex{2}$ with the usual vector addition, but with scalar multiplication defined by \begin{equation*} \alpha \colvector{x\\y} = \colvector{\alpha y \\ \alpha x} \end{equation*} Determine whether or not $V$ is a vector space with these operations.
Let $V$ be the set $\complex{2}$ with the usual scalar multiplication, but with vector addition defined by \begin{equation*} \colvector{x\\y} + \colvector{z\\w} = \colvector{y + w \\ x + z} \end{equation*} Determine whether or not $V$ is a vector space with these operations.
Let $V$ be the set $M_{22}$ with the usual scalar multiplication, but with addition defined by $A + B = \zeromatrix_{2,2}$ for all $2 \times 2$ matrices $A$ and $B$. Determine whether or not $V$ is a vector space with these operations.
Let $V$ be the set $M_{22}$ with the usual addition, but with scalar multiplication defined by $\alpha A= \zeromatrix_{2,2}$ for all $2 \times 2$ matrices $A$ and scalars $\alpha$. Determine whether or not $V$ is a vector space with these operations.
Consider the following sets of $3 \times 3$ matrices, where the symbol $*$ indicates the position of an arbitrary complex number. Determine whether or not these sets form vector spaces with the usual operations of addition and scalar multiplication for matrices.
Explain why we need to define the vector space $P_n$ as the set of all polynomials with degree up to and including $n$ instead of the more obvious set of all polynomials of degree exactly $n$.
The set of integers is denoted $\mathbb{Z}$. Does the set $\mathbb{Z}^2 = \setparts{\colvector{m\\n}}{m,n\in\mathbb{Z}}$ with the operations of standard addition and scalar multiplication of vectors form a vector space?
Prove each of the ten properties of Definition VS for each of the following examples of a vector space: Example VSP, Example VSIS, Example VSF, Example VSS.
The next three problems suggest that under the right situations we can “cancel.” In practice, these techniques should be avoided in other proofs. Prove each of the following statements.
Suppose that $V$ is a vector space, and $\vect{u},\,\vect{v},\,\vect{w}\in V$. If $\vect{w}+\vect{u}=\vect{w}+\vect{v}$, then $\vect{u}=\vect{v}$.
Suppose $V$ is a vector space, $\vect{u},\,\vect{v}\in V$ and $\alpha$ is a nonzero scalar from $\complex{\null}$. If $\alpha\vect{u}=\alpha\vect{v}$, then $\vect{u}=\vect{v}$.
Suppose $V$ is a vector space, $\vect{u}\neq\zerovector$ is a vector in $V$ and $\alpha,\,\beta\in\complex{\null}$. If $\alpha\vect{u}=\beta\vect{u}$, then $\alpha=\beta$.
Suppose that $V$ is a vector space and $\alpha\in\complexes$ is a scalar such that $\alpha\vect{x}=\vect{x}$ for every $\vect{x}\in V$. Prove that $\alpha = 1$. In other words, Property O is not duplicated for any other scalar but the “special” scalar, 1. (This question was suggested by James Gallagher.)