Riley Coleman
02/28/2023 · Primary School
5. Let \( U_{1}, U_{2}, U_{3} \) be the following subspaces of \( \mathbb{R}^{3} \) : \( U_{1}=\{(a, b, c): a=c\}, U_{2}=\{(a ; b ; c): a+b=c=0\}, U_{3}=\{(0 ; 0 ; c)\} \) Show that (a) \( \mathbf{R}^{3}=U_{1}+U_{3} \). When is the sum direct?
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To show \( \mathbb{R}^{3}=U_{1}+U_{3} \), express any vector in \( \mathbb{R}^{3} \) as a sum of a vector from \( U_{1} \) and a vector from \( U_{3} \). The sum is direct if \( U_{1} \cap U_{3} \) only contains the zero vector.
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