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

$R}^{4$

emancipezN
2021-10-31
Answered

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

$R}^{4$

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Theodore Schwartz

Answered 2021-11-01
Author has **99** answers

Every vector in $R}^{4$ can be represented as

$\upsilon =({\upsilon}_{1},{\upsilon}_{2},{\upsilon}_{3},{\upsilon}_{4})$

We have to find the additive identify vector - the vector that has following property:

$\upsilon +x=x+\upsilon =\upsilon$

In coordinate notation:

$({\upsilon}_{1},{\upsilon}_{2},{\upsilon}_{3},{\upsilon}_{4})+({x}_{1},{x}_{2},{x}_{3},{x}_{4})=({\upsilon}_{1}+{x}_{1},{\upsilon}_{2}+{x}_{2},{\upsilon}_{3}+{x}_{3},{\upsilon}_{4}+{x}_{4})=({\upsilon}_{1},{\upsilon}_{2},{\upsilon}_{3},{\upsilon}_{4})$

$\upsilon}_{1}+{x}_{1}={\upsilon}_{1$

$\upsilon}_{2}+{x}_{2}={\upsilon}_{2$

$\upsilon}_{3}+{x}_{3}={\upsilon}_{3$

$\upsilon}_{4}+{x}_{4}={\upsilon}_{4$

From properties of addition in R, we know that${x}_{1}={x}_{2}={x}_{3}={x}_{4}=0$

So, additive identity vector is x=(0,0,0,0)

We have to find the additive identify vector - the vector that has following property:

In coordinate notation:

From properties of addition in R, we know that

So, additive identity vector is x=(0,0,0,0)

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