I need help to solve this question !!
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HallsofIvy
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First, have you calculated \(\displaystyle \vec{w}\)? Second, do you see that the "additive identity" is \(\displaystyle \begin{bmatrix}-4\\ -2 \\ -3\end{bmatrix}\)?
HallsofIvy
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Well, I hope you at least know the bases or axioms of vector space. In particular every vector space has an "additive identity"or "0 vector" (typically represented by "0" but not necessarily having anything to do with the number 0) such that, for any vector, v, v+ 0= v.
Here you are defining addition by \(\displaystyle \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}+ \begin{pmatrix}v_0 \\ v_1 \\ v_2 \end{pmatrix}= \begin{pmatrix}u_0+ v_0+ 4 \\ u_1+v_1+ 2 \\ u_2+ v_2+ 3\end{pmatrix}\) so writing the additive identity (0 vector) as \(\displaystyle \begin{pmatrix} a \\ b \\ c \end{pmatrix}\) we must have\(\displaystyle \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}+ \begin{pmatrix}a \\ b \\ c \end{pmatrix}= \begin{pmatrix}u_0+ a+ 4 \\ u_1+b+ 2 \\ u_2+ c+ 3\end{pmatrix}= \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}\).
So we must have \(\displaystyle u_0+ a+ 4= u_0\), \(\displaystyle u_1+ b+ 2= u_1\), and \(\displaystyle u_2+ b+ 3= u_2\) so a= -4, b= -2, and c= -3. That is, the additive identity, or 0 vector, is \(\displaystyle \begin{pmatrix}a \\ b \\ c\end{pmatrix}= \begin{pmatrix}-4\\ -2 \\ -3 \end{pmatrix}\).
Here you are defining addition by \(\displaystyle \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}+ \begin{pmatrix}v_0 \\ v_1 \\ v_2 \end{pmatrix}= \begin{pmatrix}u_0+ v_0+ 4 \\ u_1+v_1+ 2 \\ u_2+ v_2+ 3\end{pmatrix}\) so writing the additive identity (0 vector) as \(\displaystyle \begin{pmatrix} a \\ b \\ c \end{pmatrix}\) we must have\(\displaystyle \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}+ \begin{pmatrix}a \\ b \\ c \end{pmatrix}= \begin{pmatrix}u_0+ a+ 4 \\ u_1+b+ 2 \\ u_2+ c+ 3\end{pmatrix}= \begin{pmatrix} u_0 \\ u_1\\u_2\end{pmatrix}\).
So we must have \(\displaystyle u_0+ a+ 4= u_0\), \(\displaystyle u_1+ b+ 2= u_1\), and \(\displaystyle u_2+ b+ 3= u_2\) so a= -4, b= -2, and c= -3. That is, the additive identity, or 0 vector, is \(\displaystyle \begin{pmatrix}a \\ b \\ c\end{pmatrix}= \begin{pmatrix}-4\\ -2 \\ -3 \end{pmatrix}\).
HallsofIvy
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What is the additive identity? Calling it [math]\begin(pmatrix} e_0 \\ e_1 \\ e_2\end{pmatrix}[/math] we must have $\begin(pmatrix} v_0 \\ v_1 \\ v_2\end{pmatrix}+ \begin(pmatrix} e_0 \\ e_1 \\ e_2\end{pmatrix}= \begin(pmatrix} v_0+e_0+ 4 \\ v_1+ e_1+ 3 \\ v_2+ e_2+ 2\end{pmatrix}= \begin(pmatrix} v_0 \\ v_1 \\ v_2\end{pmatrix}$
so we must have $v_0+ e_0+ 4= v_0$, $v_1+ e_1+ 3= v_1$, $v_2+ e_2+ 2= v_2$. That is, $e_0= -4$, $e_1= -3$, and $e_2= -2$ so the additive identity is $\begin{pmatrix}-4 \\ -3 \\ -2\end{pmatrix}$
so we must have $v_0+ e_0+ 4= v_0$, $v_1+ e_1+ 3= v_1$, $v_2+ e_2+ 2= v_2$. That is, $e_0= -4$, $e_1= -3$, and $e_2= -2$ so the additive identity is $\begin{pmatrix}-4 \\ -3 \\ -2\end{pmatrix}$
