Exercise · ZW-GRAM-SCHMIDT-PROCESS-005 · ★☆☆ Basic
After Gram–Schmidt, the first vector is \(w_1=(3,4)\). What is the corresponding unit vector \(q_1\)?
Choose one option.
Show answer
- C. \((\frac{3}{5},\frac{4}{5})\)
Explanation
\(\|w_1\|=5\), so \(q_1=w_1/5=(\tfrac35,\tfrac45)\). Dividing by \(\|w_1\|^2=25\) is a common mistake.
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?
- ZW-GRAM-SCHMIDT-PROCESS-002 Apply Gram–Schmidt to v_1=(2,0), v_2=(3,4). What is w_2?
- 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-006 Apply Gram–Schmidt to v_1=(1,1,0), v_2=(1,0,1). What is w_2?
- ZW-GRAM-SCHMIDT-PROCESS-007 Apply Gram–Schmidt to v_1=(1,1,1), v_2=(1,2,3). 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?