Exercise · ZW-GRAM-SCHMIDT-PROCESS-007 · ★★☆ Intermediate
Apply Gram–Schmidt to \(v_1=(1,1,1)\), \(v_2=(1,2,3)\). What is \(w_2\)?
Choose one option.
Show answer
- B. \((-1,0,1)\)
Explanation
\(\frac{v_2\cdot w_1}{w_1\cdot w_1}=\frac63=2\), so \(w_2=(1,2,3)-2(1,1,1)=(-1,0,1)\). Watch the sign: \((1,0,-1)\) is the opposite.
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
- ZW-GRAM-SCHMIDT-PROCESS-001 Apply Gram–Schmidt to v_1=(1,1), v_2=(1,0). What is the second orthogonal vector w_2?
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- ZW-GRAM-SCHMIDT-PROCESS-003 Apply Gram–Schmidt to v_1=(1,-1), v_2=(2,0). What is w_2?
- ZW-GRAM-SCHMIDT-PROCESS-004 In the Gram–Schmidt process, what is taken as the first orthogonal vector w_1?
- ZW-GRAM-SCHMIDT-PROCESS-005 After Gram–Schmidt, the first vector is w_1=(3,4). What is the corresponding unit vector q_1?
- ZW-GRAM-SCHMIDT-PROCESS-006 Apply Gram–Schmidt to v_1=(1,1,0), v_2=(1,0,1). What is w_2?
- ZW-GRAM-SCHMIDT-PROCESS-008 Apply Gram–Schmidt to v_1=(1,0,0), v_2=(1,1,0), v_3=(1,1,1). What is w_3?
- ZW-GRAM-SCHMIDT-PROCESS-009 Gram–Schmidt is applied to v_1,v_2 with v_2=3v_1 and v_1≠mathbf 0. What is w_2?