Question

If \(W\) is a subspace of a finite dimensional vector space \(V\), then \(\dim\left(\dfrac{V}{W}\right)=\)

• A. \(\dim V+\dim W\)
• B. \(\dim V-\dim W\)
• C. \(\dfrac{\dim V}{\dim W}\)
• D. \(\dfrac{\dim W}{\dim V}\)

Hint:

For quotient spaces, remember: \(\dim(V/W)=\dim V-\dim W\).

Answer 1

The Correct Option is B

Concept:
If \(W\) is a subspace of a finite dimensional vector space \(V\), then the quotient space is written as \[ \frac{V}{W} \] The dimension of the quotient space is given by \[ \dim\left(\frac{V}{W}\right)=\dim V-\dim W \]

Step 1: Understand quotient space. The quotient space \(\dfrac{V}{W}\) contains cosets of the form \[ v+W \] where \(v\in V\). It measures the part of \(V\) which remains after identifying all vectors of \(W\) as zero.

Step 2: Use dimension formula. For finite dimensional vector spaces, \[ \dim\left(\frac{V}{W}\right)=\dim V-\dim W \]

Step 3: Final answer. \[ \boxed{\dim V-\dim W} \]