 Let u and v be distinct vectors of a vector Kathleen Rausch 2022-01-07 Answered
Let u and v be distinct vectors of a vector space V. Show that if {u, v} is a basis for V and a and b are nonzero scalars, then both {u+v, au} and {au, bv} are also bases for V.

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Given, u and v be distinct vectors of a vector space V and {u, v} is a basis for V and a and b are nonzero scalars.
We have to show: both {u + v, au} and {au, bv} are also bases for V.
$$\displaystyle{u}\ne{v},{a},{b}\ne{0}$$
$$\displaystyle{\left\lbrace{u},{v}\right\rbrace}$$ is a basis for V $$\displaystyle\Rightarrow{\left\lbrace{u}+{v},{a}{u}\right\rbrace}$$ and $$\displaystyle{\left\lbrace{a}{u},{b}{v}\right\rbrace}$$ are bases for V
From basis $$\displaystyle{\left\lbrace{u},{v}\right\rbrace}$$ we get dimension of V
$$\displaystyle{\left\lbrace{u},{v}\right\rbrace}$$ is a basis for V $$\displaystyle\Rightarrow$$ V is a 2-dimensional vector space and $$\displaystyle{\left(\cdot\right)}\alpha{u}+\beta{v}={0}\Rightarrow\alpha=\beta={0}$$
Show $$\displaystyle{\left\lbrace{u}+{v},{a}{u}\right\rbrace}$$ is a basis for V:
$$\displaystyle{\left\lbrace{u}+{v},{a}{u}\right\rbrace}$$:
$$\displaystyle{A}{\left({u}+{v}\right)}+{B}{\left({a}{u}\right)}={0}\Leftrightarrow{A}{u}+{A}{v}+{B}{a}{u}={0}\Leftrightarrow$$
$$\displaystyle{\left({A}+{B}{a}\right)}{u}+{\left({B}\right)}{v}={0}$$
By the step (in second step)
$$\displaystyle{A}+{B}{a}={0},{a}\ne{0}$$
$$\displaystyle{B}={0}$$
$$\displaystyle\Rightarrow{A}={B}={0}\Rightarrow{\left\lbrace{u}+{v},{a}{u}\right\rbrace}$$ is linearly independent set of two vectors
$$\displaystyle\Rightarrow{\left\lbrace{u}+{v},{a}{u}\right\rbrace}$$ is a basis for V
(Proved)
Not exactly what you’re looking for? xandir307dc

here is the continuation of the solution:
Show $$\displaystyle{\left\lbrace{a}{u},{b}{v}\right\rbrace}$$ is a basis for V
$$\displaystyle{\left\lbrace{a}{u},{b}{v}\right\rbrace}:$$
$$A(au)+B(bv)=0 \Leftrightarrow (Aa)u+(bb)v=0$$
By the step (in second step)
$$\displaystyle{A}{a}={0},{a}\ne{0}$$
$$\displaystyle{B}{b}={0},{b}\ne{0}$$
$$\displaystyle\Rightarrow{A}={B}={0}\Rightarrow{\left\lbrace{a}{u},{b}{v}\right\rbrace}:$$ is linearly independent set of two vectors
$$\displaystyle\Rightarrow{\left\lbrace{a}{u},{b}{v}\right\rbrace}:$$ is a basis for V
(Proved)