Question

Given a real subspace \( W \) of \( {R}^4 \), let \( W^\perp \) denote its orthogonal complement with respect to the standard inner product on \( {R}^4 \). Let \( W_1 = {Span}\{(1, 0, 0, -1)\ \) and \( W_2 = {Span}\{(2, 1, 0, -1)\} \). The dimension of \( W_1^\perp \cap W_2^\perp \) over \( {R} \) is equal to (answer in integer):}

Hint:

For orthogonal complements, use the formula \( \dim(W_1 \cap W_2) = \dim(W_1) + \dim(W_2) - \dim({R}^n) \).

Answer 1

Step 1: Orthogonal complement properties. The subspaces \( W_1^\perp \) and \( W_2^\perp \) intersect in a subspace of \( {R}^4 \). Step 2: Computing dimensions. The vectors defining \( W_1 \) and \( W_2 \) span a subspace of dimension 2, leaving a 2-dimensional intersection in \( {R}^4 \). Step 3: Conclusion. The dimension of \( W_1^\perp \cap W_2^\perp \) is \( {2} \).