Exercise · ZW-GRAM-SCHMIDT-PROCESS-010 · ★★★ Advanced

Which statement about \(w_1,w_2\), the output of Gram–Schmidt applied to linearly independent \(v_1,v_2\), is true?

Choose one option.

Show answer

B. They are orthogonal and span the same as \(v_1,v_2\)

Explanation

Since \(w_1=v_1\) and \(w_2=v_2-c\,v_1\), each pair comes from the other by linear combinations, so they span the same space. They are orthogonal, but unit only after dividing by their norms.

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Review the concept: Gram–Schmidt process

A procedure that turns linearly independent vectors into an orthogonal list spanning the same subspace, by subtracting from each one its projections onto the previous ones.

\[w_k=v_k-\sum_{j

More practice problems: gram–schmidt process

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