**– 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**.

**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**:

\[R+I=I+R=R\]

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**.

## 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:

\[(I_1,\ {\ I}_2)\ =\ (0,\ 0)\]

\[\mathbb{I}^2\ =\ (0,\ 0)\]