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Problem 11. Let A € Rnxm. For each 1 ≤ l ≤ m let e € Rmx1 be the vector whose lth component is 1 and all other entries are zero. For example, if m = 3 then 0 e₁ = 0 0 = " A has n rows and m columns, let de € Rn be the lth column vector of A so that | | A = ā₁ āz āz and €3 = Hint: First express as = 21 Xili. (a) For 1 ≤ ≤m, what is Aeg? Hint: Consider (Ael)ij (b) Let ☞ € R™ be x = (x1, x2, ... Xm). Using your answer from (a) show that Az = ām m =Σxiai. i=1

Problem 11. Let A € Rnxm. For each 1 ≤ l ≤ m let e € Rmx1 be the vector whose lth component is 1 and all other entries are zero. For example, if m = 3 then 0 e₁ = 0 0 = " A has n rows and m columns, let de € Rn be the lth column vector of A so that | | A = ā₁ āz āz and €3 = Hint: First express as = 21 Xili. (a) For 1 ≤ ≤m, what is Aeg? Hint: Consider (Ael)ij (b) Let ☞ € R™ be x = (x1, x2, ... Xm). Using your answer from (a) show that Az = ām m =Σxiai. i=1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 45E
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Problem 11. Let A € Rn×m. For each 1 ≤ l ≤ m let e € Rm×¹ be the vector whose lth component is 1 and all other entries are zero. For example, if m = 3 then e₁ 日 9 e2 (a) For 1 ≤ l < m, what is Aee? Hint: Consider (Ael)ij (b) Let = Rm be x = (x1, x2, = A and A has n rows and m columns, let ā, E Rn be the lth column vector of A so that A 1- [4 44 - 4 = ām m m Hint: First express as = 1 xili. e3 .ïm)¹. Using your answer from (a) show that Ax = Σxidi. i=1 = 0
Transcribed Image Text:Problem 11. Let A € Rn×m. For each 1 ≤ l ≤ m let e € Rm×¹ be the vector whose lth component is 1 and all other entries are zero. For example, if m = 3 then e₁ 日 9 e2 (a) For 1 ≤ l < m, what is Aee? Hint: Consider (Ael)ij (b) Let = Rm be x = (x1, x2, = A and A has n rows and m columns, let ā, E Rn be the lth column vector of A so that A 1- [4 44 - 4 = ām m m Hint: First express as = 1 xili. e3 .ïm)¹. Using your answer from (a) show that Ax = Σxidi. i=1 = 0
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