– Given Vector Space:
\[\mathbb{R}^4\]
The aim of this article is to find the Zero Vector for the given vector space. The basic concept behind this article is the Additive Identity of a Vector Space.
Real number
Additive Identity is defined as the value that if added or subtracted from a second value, does not change it. For example, if we add $0$ to any real numbers, it does not change the value of the given real numbers. We can call Zero $0$ the Additive Identity of the Real Numbers.
If we consider $R$ as a real number and $I$ as an Additive Identity, then as per Additive Identity Law:
\[R+I=I+R=R\]
Subtraction real number
A Vector Space is defined as a Set comprising of one or more vector elements and it is represented by $\mathbb{R}^n$ where $n$ represents the number of elements in the given vector space.
Addition real number
Expert Answer
Given that:
Vector Space $=\mathbb{R}^4$
This shows that $\mathbb{R}^4$ has $4$ vector elements.
Let us represent $\mathbb{R}^4$ as follows:
\[\mathbb{R}^4 =\ (R_1,\ R_2,\ R_3,\ R_4)\]
Let us suppose that:
Additive Identity $=\mathbb{I}^4$
Let us represent $= \mathbb{I}^4$ as follows:
\[\mathbb{I}^4 = (I_1,\ I_2,\ I_3,\ I_4)\]
As per Additive Identity Law:
\[\mathbb{R}^4\ +\mathbb{I}^4\ =\mathbb{I}^4\ +\mathbb{R}^4\ =\ \mathbb{R}^4\]
Substituting the values:
\[(R_1,\ R_2,\ R_3,\ R_4)\ +\ (I_1,\ I_2,\ I_3,\ I_4)\ =\ (R_1,\ R_2,\ R_3,\ R_4)\]
Performing addition of vector elements:
\[(R_1\ +\ I_1,\ R_2\ +{\ I}_2,\ R_3\ +{\ I}_3,\ R_4{\ +\ I}_4)\ =\ (R_1,\ R_2,\ R_3,\ R_4)\]
Comparing element by element:
First Element:
\[R_1\ +{\ I}_1\ =\ R_1\]
\[I_1\ =\ R_1\ -{\ R}_1\]
\[I_1\ =\ 0\]
Second Element:
\[R_2\ +\ I_2\ ={\ R}_2\]
\[I_2\ ={\ R}_2\ -{\ R}_2\]
\[I_2\ =\ 0\]
Third Element:
\[R_3\ +\ I_3\ =\ R_3\]
\[I_3\ =\ R_3\ -\ R_3\]
\[I_3\ =\ 0\]
Fourth Element:
\[R_4\ +\ I_4\ ={\ R}_4\]
\[I_4\ =\ R_4\ -\ R_4\]
\[I_4\ =\ 0\]
Hence from the above equations, it is proved that the Additive Identity is as follows:
\[(I_1,\ I_2,\ I_3,\ I_4)\ =\ (0,\ 0,\ 0,\ 0)\]
\[\mathbb{I}^4\ =\ (0,\ 0,\ 0,\ 0)\]
Numerical Result
The Additive Identity or Zero Vector $\mathbb{I}^4$ of $\mathbb{R}^4$ is:
\[\mathbb{I}^4\ =\ (0,\ 0,\ 0,\ 0)\]
Example
For the given vector space $\mathbb{R}^2$, find the zero vector or additive identity.
Solution
Given that:
Vector Space $= \mathbb{R}^2$
This shows that $\mathbb{R}^2$ has $2$ vector elements.
Let us represent $\mathbb{R}^2$ as follows:
\[\mathbb{R}^2\ =\ (R_1,\ R_2)\]
Let us suppose that:
Additive Identity $= \mathbb{I}^2$
Let us represent $= \mathbb{I}^2$ as follows:
\[\mathbb{I}^2\ =\ (I_1,\ I_2)\]
As per Additive Identity Law:
\[\mathbb{R}^2\ +\ \mathbb{I}^2\ =\ \mathbb{I}^2\ +\ \mathbb{R}^2\ =\ \mathbb{R}^2\]
Substituting the values:
\[(R_1,\ {\ R}_2)\ +\ (I_1,\ \ I_2)\ =\ (R_1,\ R_2)\]
Performing addition of vector elements:
\[(R_1\ +{\ I}_1,\ \ R_2\ +\ I_2)\ =\ (R_1,\ R_2)\]
Comparing element by element:
First Element:
\[R_1\ +{\ I}_1\ =\ {\ R}_1\]
\[I_1\ ={\ R}_1\ -{\ R}_1\]
\[I_1\ =\ 0\]
Second Element:
\[R_2\ +\ I_2\ ={\ R}_2\]
\[I_2\ ={\ R}_2\ -{\ R}_2\]
\[I_2\ =\ 0\]
Hence from the above equations, it is proved that the Additive Identity is as follows:
\[(I_1,\ {\ I}_2)\ =\ (0,\ 0)\]
\[\mathbb{I}^2\ =\ (0,\ 0)\]