Exam-style question
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Let u = [2, 0] and v = [0, 4]. Determine whether 0.25u + 0.75v is a convex combination of u and v. Calculate the resulting vector and justify your answer.
Model answer
What a good answer should say
- It is a convex combination because both coefficients are non-negative and their sum is 0.25 + 0.75 = 1.
- The resulting vector is 0.25[2, 0] + 0.75[0, 4] = [0.5, 0] + [0, 3] = [0.5, 3].
Explanation
Why this works
A convex combination has the form αu + βv, where α and β are both at least zero and α + β equals 1. The given coefficients satisfy these conditions, and the vector is found by scaling each vector before adding corresponding entries.
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