Question

If $W_1$ and $W_2$ are finite dimensional subspaces of a vector space $V$, then:

• A. $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2)$
• B. $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) + \dim(W_1 \cap W_2)$
• C. $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$
• D. $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) + \dim(W_1 \cup W_2)$

Hint:

This formula is analogous to the Inclusion-Exclusion Principle for sets: $|A \cup B| = |A| + |B| - |A \cap B|$. If $W_1 \cap W_2 = \{0\}$ (direct sum), then $\dim(W_1 \oplus W_2) = \dim(W_1) + \dim(W_2)$.

Answer 1

The Correct Option is C

Step 1: Concept
In vector space theory, the sum of two subspaces $W_1$ and $W_2$ is defined as: \[ W_1 + W_2 = \{ w_1 + w_2 \mid w_1 \in W_1, w_2 \in W_2 \} \] Both $W_1 + W_2$ and $W_1 \cap W_2$ are subspaces of $V$. The relationship between their dimensions is given by the Second Isomorphism Theorem for Vector Spaces (often called Grassmann's Dimension Formula).

Step 2: Key Formulas and Approach
Grassmann's Dimension Theorem: For any two finite-dimensional subspaces $W_1$ and $W_2$ of a vector space $V$: \[ \dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2) \]

Step 3: Step-by-step Explanation

• Let $\dim(W_1 \cap W_2) = k$. Choose a basis $\{v_1, v_2, \dots, v_k\}$ for $W_1 \cap W_2$.

• By the Basis Extension Theorem, extend this basis to a basis of $W_1$: \[ \mathcal{B}_1 = \{v_1, \dots, v_k, u_1, \dots, u_m\} \implies \dim(W_1) = k + m \]
• Similarly, extend the basis of $W_1 \cap W_2$ to a basis of $W_2$: \[ \mathcal{B}_2 = \{v_1, \dots, v_k, w_1, \dots, w_n\} \implies \dim(W_2) = k + n \]
• Then the set $\mathcal{B} = \{v_1, \dots, v_k, u_1, \dots, u_m, w_1, \dots, w_n\}$ spans $W_1 + W_2$ and is linearly independent, so it forms a basis of $W_1 + W_2$.

• Therefore, the dimension of $W_1 + W_2$ is: \[ \dim(W_1 + W_2) = k + m + n \]
• Expressing this in terms of individual dimensions: \[ \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2) = (k + m) + (k + n) - k = k + m + n \]
• Thus, $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$.

Step 4: Final Answer
Grassmann's dimension formula states that $\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2)$. Thus, Option (C) is correct.