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Mathematics
Linear and Abstract Algebra
Prove the statements : Vectors/Matrices
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[QUOTE="mathmari, post: 6777170, member: 558163"] We define addition in $\mathbb{R}^n$ componentwise: $x+y=(x_1, \ldots , x_n)+(y_1, \ldots , y_n)=(x_1+y_1, \ldots , x_n+y_n)$. We have that $0_{\mathbb{R}^n}=(0,\ldots , 0)$. So we have that \begin{equation*} v+0_{\mathbb{R}^n}=(v_1, \ldots , v_n)+(0,\ldots , 0)=(v_1+0, \ldots , v_n+0)=(v_1, \ldots , v_n)=v\end{equation*} Similarily, we get \begin{equation*}0_{\mathbb{R}^n}+v=(0,\ldots , 0)+(v_1, \ldots , v_n)=(0+v_1, \ldots , 0+v_n)=(v_1, \ldots , v_n)=v\end{equation*} Therefore, it holds that $v+0_{\mathbb{R}^n}=v=0_{\mathbb{R}^n}+v$. The scalar multiplication is defined as follows: $\lambda x=\lambda (x_1, \ldots , x_n)=(\lambda x_1, \ldots , \lambda x_n)$. Then we get \begin{align*}\lambda (v+w)&=\lambda \left ((v_1, \ldots , v_n)+(w_1, \ldots , w_n)\right )=\lambda \left ((v_1+w_1, \ldots , v_n+w_n)\right ) \\ & =(\lambda(v_1+w_1), \ldots , \lambda(v_n+w_n))=(\lambda v_1+\lambda w_1, \ldots , \lambda v_n+\lambda w_n) \\ & =(\lambda v_1, \ldots , \lambda v_n)+(\lambda w_1, \ldots , \lambda w_n) =\lambda( v_1, \ldots , v_n)+ \lambda (w_1, \ldots , w_n) \\ & =\lambda v+\lambda w\end{align*} Are these proofs correct and complete? (Wondering) [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Prove the statements : Vectors/Matrices
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