Let 𝒱 be a vector space over 𝔽. Show that if α, β∈𝔽 and if 𝐯 is a nonzero vector in 𝒱, then α𝐯=β

step 1 Okay, as per the question, we have to concern a vector space v, a basis, B, that is B1, B2, and

Let 𝒱 be a vector space over 𝔽. Show that if α, β∈𝔽 and if 𝐯...

Let $\mathcal{V}$ be a vector space over $\mathbb{F}$. Show that if $\alpha, \beta \in \mathbb{F}$ and if $\mathbf{v}$ is a nonzero vector in $\mathcal{V}$, then $\alpha \mathbf{v}=\beta \mathbf{v} \Longleftrightarrow \alpha=\beta$. [HINT: $\left.\alpha-\beta \neq 0 \Longrightarrow \mathbf{v}=(\alpha-\beta)^{-1}(\alpha-\beta) \mathbf{v}\right)$.] Example 1.1. The set of column vectors $$ \mathbb{F}^p=\left\{\left[\begin{array}{c} x_1 \\ \vdots \\ x_p \end{array}\right]: x_i \in \mathbb{F}, i=1, \ldots, p\right\} $$ of height $p$ with entries $x_i \in \mathbb{F}$ that are subject to the natural rules of vector addition $$ \left[\begin{array}{c} x_1 \\ \vdots \\ x_p \end{array}\right]+\left[\begin{array}{c} y_1 \\ \vdots \\ y_p \end{array}\right]=\left[\begin{array}{c} x_1+y_1 \\ \vdots \\ x_p+y_p \end{array}\right] $$ and multiplication $$ \alpha\left[\begin{array}{c} x_1 \\ \vdots \\ x_p \end{array}\right]=\left[\begin{array}{c} \alpha x_1 \\ \vdots \\ \alpha x_p \end{array}\right] $$ of the vector $\mathbf{x}$ by a number $\alpha \in \mathbb{F}$ is the most basic example of a vector space. Note the difference between the number 0 and the vector $0 \in \mathbb{F}^p$. The latter is a column vector of height $p$ with all $p$ entries equal to the number zero. The set $\mathbb{F}^{p \times q}$ of $p \times q$ matrices with entries in $\mathbb{F}$ is a vector space with respect to the rules of vector addition: $$ \left[\begin{array}{ccc} x_{11} & \cdots & x_{1 q} \\ \vdots & & \vdots \\ x_{p 1} & \cdots & x_{p q} \end{array}\right]+\left[\begin{array}{ccc} y_{11} & \cdots & y_{1 q} \\ \vdots & & \vdots \\ y_{p 1} & \cdots & y_{p q} \end{array}\right]=\left[\begin{array}{ccc} x_{11}+y_{11} & \cdots & x_{1 q}+y_{1 q} \\ \vdots & & \vdots \\ x_{p 1}+y_{p 1} & \cdots & x_{p q}+y_{p q} \end{array}\right], $$ and multiplication by a scalar $\alpha \in \mathbb{F}$ : $$ \alpha\left[\begin{array}{ccc} x_{11} & \cdots & x_{1 q} \\ \vdots & & \vdots \\ x_{p 1} & \cdots & x_{p q} \end{array}\right]=\left[\begin{array}{ccc} \alpha x_{11} & \cdots & \alpha x_{1 q} \\ \vdots & & \vdots \\ \alpha x_{p 1} & \cdots & \alpha x_{p q} \end{array}\right] . $$ Notice that the vector space $\mathbb{F}^p$ dealt with a little earlier coincides with the vector space that is designated $\mathbb{F}^{p \times 1}$ in the current example.

Let $\mathcal{V}$ be a vector space over $\mathbb{F}$. Show that if $\alpha, \beta \in \mathbb{F}$ and if $\mathbf{v}$ is a nonzero vector in $\mathcal{V}$, then $\alpha \mathbf{v}=\beta \mathbf{v} \Longleftrightarrow \alpha=\beta$. [HINT: $\left.\alpha-\beta \neq 0 \Longrightarrow \mathbf{v}=(\alpha-\beta)^{-1}(\alpha-\beta) \mathbf{v}\right)$.]

Example 1.1. The set of column vectors
$$
\mathbb{F}^p=\left\{\left[\begin{array}{c}
x_1 \\
\vdots \\
x_p
\end{array}\right]: x_i \in \mathbb{F}, i=1, \ldots, p\right\}
$$
of height $p$ with entries $x_i \in \mathbb{F}$ that are subject to the natural rules of vector addition
$$
\left[\begin{array}{c}
x_1 \\
\vdots \\
x_p
\end{array}\right]+\left[\begin{array}{c}
y_1 \\
\vdots \\
y_p
\end{array}\right]=\left[\begin{array}{c}
x_1+y_1 \\
\vdots \\
x_p+y_p
\end{array}\right]
$$
and multiplication
$$
\alpha\left[\begin{array}{c}
x_1 \\
\vdots \\
x_p
\end{array}\right]=\left[\begin{array}{c}
\alpha x_1 \\
\vdots \\
\alpha x_p
\end{array}\right]
$$
of the vector $\mathbf{x}$ by a number $\alpha \in \mathbb{F}$ is the most basic example of a vector space. Note the difference between the number 0 and the vector $0 \in \mathbb{F}^p$. The latter is a column vector of height $p$ with all $p$ entries equal to the number zero.
The set $\mathbb{F}^{p \times q}$ of $p \times q$ matrices with entries in $\mathbb{F}$ is a vector space with respect to the rules of vector addition:
$$
\left[\begin{array}{ccc}
x_{11} & \cdots & x_{1 q} \\
\vdots & & \vdots \\
x_{p 1} & \cdots & x_{p q}
\end{array}\right]+\left[\begin{array}{ccc}
y_{11} & \cdots & y_{1 q} \\
\vdots & & \vdots \\
y_{p 1} & \cdots & y_{p q}
\end{array}\right]=\left[\begin{array}{ccc}
x_{11}+y_{11} & \cdots & x_{1 q}+y_{1 q} \\
\vdots & & \vdots \\
x_{p 1}+y_{p 1} & \cdots & x_{p q}+y_{p q}
\end{array}\right],
$$
and multiplication by a scalar $\alpha \in \mathbb{F}$ :
$$
\alpha\left[\begin{array}{ccc}
x_{11} & \cdots & x_{1 q} \\
\vdots & & \vdots \\
x_{p 1} & \cdots & x_{p q}
\end{array}\right]=\left[\begin{array}{ccc}
\alpha x_{11} & \cdots & \alpha x_{1 q} \\
\vdots & & \vdots \\
\alpha x_{p 1} & \cdots & \alpha x_{p q}
\end{array}\right] .
$$

Notice that the vector space $\mathbb{F}^p$ dealt with a little earlier coincides with the vector space that is designated $\mathbb{F}^{p \times 1}$ in the current example.

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