Question:
If \(W\) is a subspace of \(V^3\), where
\[
W=\{(a,b,c)\mid a+b+c=0\},
\]
then \(\dim W=\)
If \(W\) is a subspace of \(V^3\), where
\[
W=\{(a,b,c)\mid a+b+c=0\},
\]
then \(\dim W=\)
Show Hint
One independent linear equation in \(V^3\) reduces the dimension from \(3\) to \(2\).
Updated On: Jul 4, 2026
- \(2\)
- \(3\)
- \(1\)
- \(0\)
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
The dimension of a subspace is the number of free parameters needed to describe its vectors. Given, \[ W=\{(a,b,c):a+b+c=0\} \] This is a subspace of \(V^3\).
Step 1: Use the condition. \[ a+b+c=0 \] So, \[ c=-a-b \]
Step 2: Write general vector of \(W\). \[ (a,b,c)=(a,b,-a-b) \] \[ (a,b,-a-b)=a(1,0,-1)+b(0,1,-1) \]
Step 3: Count independent parameters. The vector depends on two arbitrary parameters: \[ a,\quad b \] Therefore, \[ \dim W=2 \]
Step 4: Final answer. \[ \boxed{2} \]
The dimension of a subspace is the number of free parameters needed to describe its vectors. Given, \[ W=\{(a,b,c):a+b+c=0\} \] This is a subspace of \(V^3\).
Step 1: Use the condition. \[ a+b+c=0 \] So, \[ c=-a-b \]
Step 2: Write general vector of \(W\). \[ (a,b,c)=(a,b,-a-b) \] \[ (a,b,-a-b)=a(1,0,-1)+b(0,1,-1) \]
Step 3: Count independent parameters. The vector depends on two arbitrary parameters: \[ a,\quad b \] Therefore, \[ \dim W=2 \]
Step 4: Final answer. \[ \boxed{2} \]
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