Describe the zero vector (the additive identity) of the vector

Mahagnazk 2021-11-11 Answered
Describe the zero vector (the additive identity) of the vector space.
\(\displaystyle{R}^{{4}}\)

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Expert Answer

Knes1997
Answered 2021-11-12 Author has 7534 answers
Every vector in \(\displaystyle{\mathbb{{{R}^{{4}}}}}\) can be represented as
\(\displaystyle{v}={\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}\)
We have to find the additive identity vector - the vector that has following property:
\(\displaystyle{v}+{x}={x}+{v}={v}\)
In coordinate notation:
\(\displaystyle{\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{3}},{v}_{{4}}\right)}\)
We have to find the additive identity vector - the vector that has following property:
\(\displaystyle{v}+{x}={x}+{v}={v}\)
In coordinate notation:
\(\displaystyle{\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}+{\left({x}_{{1}},{x}_{{2}},{x}_{{3}},{x}_{{4}}\right)}={\left({v}_{{1}}+{x}_{{1}},{v}_{{2}}+{x}_{{2}},{v}_{{3}}+{x}_{{3}},{v}_{{4}}+{x}_{{4}}\right)}={\left({v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}\right)}\)
\(\displaystyle{v}_{{1}}+{x}_{{1}}={v}_{{1}}\)
\(\displaystyle{v}_{{2}}+{x}_{{2}}={v}_{{2}}\)
\(\displaystyle{v}_{{3}}+{x}_{{3}}={v}_{{3}}\)
\(\displaystyle{v}_{{4}}+{x}_{{4}}={v}_{{4}}\)
From properties of addition in \(\displaystyle{\mathbb{{{R}}}}\), we know that \(\displaystyle{x}_{{1}}={x}_{{2}}={x}_{{3}}={x}_{{4}}={0}\)
So, additive identity vector is \(\displaystyle{x}={\left({0},{0},{0},{0}\right)}\).
Result: (0,0,0,0)
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