Word Count : 582

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1) W की उपसमष्टि होने का प्रमाण और U का आधार: दिया गया है: $$ W = \{(x,4x,3x) \in \mathbb{R}^3 \mid x \in \mathbb{R} \}. $$ (i) उपसमष्टि की जाँच: * शून्य सदस्यों की उपस्थिति: $x = 0$ पर $(0,0,0) \in W$। ✔ * बंदता समिश्रण पर: यदि $\mathbf{u} = (x,4x,3x)$ और $\mathbf{v} = (y,4y,3y)$ दोनों $W$ में हैं, तो $\mathbf{u}+\mathbf{v} = (x+y,4(x+y),3(x+y)) \in W$। ✔ * बंदता स्केलर गुणा पर: यदि $\mathbf{u} = (x,4x,3x)$ और कोई स्केलर $c \in \mathbb{R}$ हो, तो $c\mathbf{u} = (cx, 4cx, 3cx) \in W$। ✔ इसलिए, $W$ $\mathbb{R}^3$ का एक उपसमष्टि है। (ii) W का आधार: सभी सदस्यों को $x$ से व्यक्त किया जा सकता है: $(x,4x,3x) = x(1,4,3)$। इसलिए $$ \text{basis}(W) = \{(1,4,3)\}. $$ (iii) $\mathbb{R}^3$ के लिए U का आधार हमें चाहिए $U$ ताकि $W \oplus U = \mathbb{R}^3$। इसका अर्थ है $W \cap U = \{0\}$ ______ ____ _______ ________ ____ _____.
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