Exercise · ZW-LINEAR-COMBINATION-AND-SPAN-009 · ★★★ Advanced
For which vectors \((x,y)\) does \(a(1,1)+b(1,2)=(x,y)\) have a solution?
Choose one option.
Show answer
- B. For every \((x,y)\in\mathbb{R}^2\)
Explanation
Solving gives \(b=y-x\) and \(a=2x-y\) with no restriction, so \(\operatorname{span}\{(1,1),(1,2)\}=\mathbb{R}^2\).
Review the concept: Linear combination and span
A linear combination of several vectors is a sum of scalar multiples of them; the span is the set of all possible linear combinations.
\[c_1v_1+\cdots+c_kv_k,\quad \operatorname{span}\{v_1,\dots,v_k\}\]
More practice problems: linear combination and span
- ZW-LINEAR-COMBINATION-AND-SPAN-001 Compute 2(1,2)+3(0,-1).
- ZW-LINEAR-COMBINATION-AND-SPAN-002 With which coefficients is (5,-2) written as c_1(1,0)+c_2(0,1)?
- ZW-LINEAR-COMBINATION-AND-SPAN-003 In ℝ^3, what set is span\(1,0,0),(0,1,0)\?
- ZW-LINEAR-COMBINATION-AND-SPAN-004 If vne 0 is in ℝ^2, what is span\v\?
- ZW-LINEAR-COMBINATION-AND-SPAN-005 Find a,b such that (4,5)=a(1,1)+b(1,-1).
- ZW-LINEAR-COMBINATION-AND-SPAN-006 Which of these vectors belongs to span\(1,0,1),(0,1,1)\?
- ZW-LINEAR-COMBINATION-AND-SPAN-007 For which value of k is the vector (2,5,k) in span\(1,2,3),(0,1,1)\?
- ZW-LINEAR-COMBINATION-AND-SPAN-008 What is span\(1,2),(2,4)\ in ℝ^2?
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