Exercise · ZW-GRAM-SCHMIDT-PROCESS-009 · ★★★ Advanced
Gram–Schmidt is applied to \(v_1,v_2\) with \(v_2=3v_1\) and \(v_1\neq\mathbf 0\). What is \(w_2\)?
Choose one option.
Show answer
- C. \(w_2=\mathbf 0\): the vectors are dependent
Explanation
\(\operatorname{proj}_{w_1}v_2=3v_1=v_2\), so \(w_2=v_2-v_2=\mathbf 0\). A zero result means \(v_2\) already lay in the span and must be discarded.
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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- ZW-GRAM-SCHMIDT-PROCESS-004 In the Gram–Schmidt process, what is taken as the first orthogonal vector w_1?
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- ZW-GRAM-SCHMIDT-PROCESS-006 Apply Gram–Schmidt to v_1=(1,1,0), v_2=(1,0,1). What is w_2?
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