Let e1=[1 0] and e2=[0 1]. show that Span {e1,e2} = R^2.

Question

Let [latex]\vec{e_{1}}=\left [ \begin{matrix} 1 \ 0 \end{matrix} ight ][/latex] and [latex]\vec{e_2}=\left [ \begin{matrix} 0 \ 1 \end{matrix} ight ][/latex]. Show that Span [latex]\left\{\vec{e_{1}},\vec{e_{2}} ight\}=\mathbb{R} ^{2}[/latex] .

Accepted Answer

We need to show that every vector in [latex]\mathbb{R} ^{2}[/latex] can be written as a linear combination of the vectors [latex]\vec{e_{1}}[/latex] and [latex]\vec{e_{2}}[/latex]. We pick a general vector [latex]\vec{x}=\left [ \begin{matrix} x_{1} \ x_{2} \end{matrix} ight ] in\ \ \mathbb{R} ^{2}[/latex] We need to determine whether there exists [latex]c_{1} , c{2} ∈ \mathbb{R} [/latex] such that

[latex]\left [ \begin{matrix} x_{1} \ x_{2} \end{matrix} ight ]=c_{1}\left [ \begin{matrix} 1 \ 0 \end{matrix} ight ] +c_{2}\left [ \begin{matrix} 0 \ 1 \end{matrix} ight ] [/latex]

We observe that we can take [latex]c_{1}=x_{1}[/latex] and [latex]c_{2}=x_{2}[/latex] That is, we have

[latex]\left [ \begin{matrix} x_{1} \ x_{2} \end{matrix} ight ]=x_{1}\left [ \begin{matrix} 1 \ 0 \end{matrix} ight ] +x_{2}\left [ \begin{matrix} 0 \ 1 \end{matrix} ight ] [/latex]

So, Span [latex]\left\{\vec{e_{1}},\vec{e_{2}} ight\}=\mathbb{R} ^{2} [/latex].

Question 1.2.3: Let e1=[1 0] and e2=[0 1]. show that Span {e1,e2} = R^2.

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