Describe the vector space's zero vector (the additive identity).R4

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Describe the vector space&#039;s zero vector (the additive identity).R4

Knes1997 · Accepted Answer

So,v=(v1,v2,v3,v4)&amp;nbsp;Find the vector that has following property:&amp;nbsp;v+x=x+v=v&amp;nbsp;(v1,v2,v3,v3,v4)&amp;nbsp;Find the vector that has following property:&amp;nbsp;v+x=x+v=v&amp;nbsp;&amp;nbsp;(v1,v2,v3,v4)+(x1,x2,x3,x4)=(v1+x1,v2+x2,v3+x3,v4+x4)=(v1,v2,v3,v4)&amp;nbsp;v1+x1=v1&amp;nbsp;v2+x2=v2&amp;nbsp;v3+x3=v3&amp;nbsp;v4+x4=v4&amp;nbsp;From properties of addition in&amp;nbsp;R, we know that&amp;nbsp;x1=x2=x3=x4=0&amp;nbsp;So, additive identity vector is&amp;nbsp;x=(0,0,0,0).&amp;nbsp;Result: (0,0,0,0)

nick1337 · Accepted Answer

In the vector space ℝ4, the zero vector, denoted as 0, is a special vector that plays a crucial role in vector arithmetic. It has the property that when added to any vector 𝐯∈ℝ4, it does not change the vector. More formally, for any vector 𝐯=[v1v2v3v4]∈ℝ4, we have:𝐯+0=[v1v2v3v4]+[0000]=[v1+0v2+0v3+0v4+0]=[v1v2v3v4]=𝐯As shown above, adding the zero vector to any vector in ℝ4 results in the original vector itself. Geometrically, the zero vector represents the origin of the coordinate system in ℝ4.To further understand the properties of the zero vector, let&#039;s consider some operations involving it. Firstly, we can subtract a vector from itself:𝐯−𝐯=[v1v2v3v4]−[v1v2v3v4]=[v1−v1v2−v2v3−v3v4−v4]=[0000]=0Subtracting a vector from itself results in the zero vector. Secondly, we can multiply the zero vector by any scalar value:k·0=k·[0000]=[k·0k·0k·0k·0]=[0000]=0Multiplying the zero vector by any scalar yields the zero vector itself. These properties showcase the unique role of the zero vector in vector spaces.Furthermore, the zero vector has another important property related to vector addition. Consider two vectors 𝐮=[u1u2u3u4] and 𝐯=[v1v2v3v4] in ℝ4. The sum of 𝐮 and bfv is given by:𝐮+𝐯=[u1u2u3u4]+[v1v2v3v4]=[u1+v1u2+v2u3+v3u4+v4]If we introduce the negative of a vector, denoted as −𝐯, which is obtained by multiplying 𝐯 by −1, we can rewrite the sum of 𝐮 and 𝐯 as follows:𝐮+𝐯=[u1u2u3u4]+[v1v2v3v4]=[u1+v1u2+v2u3+v3u4+v4]=[u1+v1u2+v2u3+v3u4+v4]+[0000]=[u1+v1+0u2+v2+0u3+v3+0u4+v4+0]=[u1+v1+(−v1+v1)u2+v2+(−v2+v2)u3+v3+(−v3+v3)u4+v4+(−v4+v4)]=[(u1+(−v1))+v1(u2+(−v2))+v2(u3+(−v3))+v3(u4+(−v4))+v4]We can observe that (−𝐯)+𝐯 results in the zero vector:(−𝐯)+𝐯=[(−v1)+v1(−v2)+v2(−v3)+v3(−v4)+v4]=[0000]=0This property implies that the negative of a vector cancels out the original vector when added together, resulting in the zero vector.In summary, the zero vector in ℝ4 is a special vector that does not change other vectors when added to them. It represents the origin of the coordinate system in

Don Sumner · Accepted Answer

Step 1. 0 is an element of R4: 0∈R4.Step 2. Addition with 0 leaves a vector unchanged: For any vector 𝐯=[v1v2v3v4]∈R4, we have 𝐯+0=[v1v2v3v4]. In other words, adding 0 to any vector 𝐯 does not change its value.Step 3. Scalar multiplication with 0 gives the zero vector: For any scalar c∈R and vector 𝐯∈R4, we have c·0=0. Multiplying the zero vector 0 by any scalar c results in the zero vector again.These properties make 0 an essential element in vector space operations and ensure that R4 is a vector space.

RizerMix · Accepted Answer

0=[0000]Here, the 0 represents the zero vector, and the [] notation indicates a column vector. The vector has four rows, where each row corresponds to one of the components of the vector. In this case, all the components are zero, so we have four zeros in the column vector.The zero vector 0 is the additive identity in the vector space ℝ4, which means that when it is added to any vector in ℝ4, it leaves the vector unchanged. For example, if we have a vector 𝐯=[v1v2v3v4] in ℝ4, adding the zero vector to 𝐯 yields the same vector:𝐯+0=[v1v2v3v4]+[0000]=[v1+0v2+0v3+0v4+0]=[v1v2v3v4]=𝐯Thus, the zero vector 0 in ℝ4 is the vector that does not change the value of any other vector when added to it.

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

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