(i.) Suppose u € R" and 7 € R™ are non-zero (column) vectors. Show thatthe n x m matrix uT has rank 1.Hint: Compare columns in the matrix v¹.(ii.) Consider the elementary basis vectors & R" (column vectors). Showthat the matrix e₁e₁ +₂e²+...+eker has rank k. Comment on thecase k = n.(iii.) Suppose ₁, ₂,..., k € R¹ are linearly independent. Similarly to theprevious question, it follows that the matrix ú₁ +₂₂+...+uku!has rank k. For the cases k = 1 and k=2, explicitly establish this forthe (linearly independent) vectors ₁ = (1,0, 2), ₂ = (1,0,−1)² € R³.

Answered: (i.) Suppose û € R¹ and 7 € R™ are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

(i.) Suppose u € R" and 7 € R™ are non-zero (column) vectors. Show that
the n x m matrix uT has rank 1.
Hint: Compare columns in the matrix v¹.
(ii.) Consider the elementary basis vectors & R" (column vectors). Show
that the matrix e₁e₁ +₂e²+...+eker has rank k. Comment on the
case k = n.
(iii.) Suppose ₁, ₂,..., k € R¹ are linearly independent. Similarly to the
previous question, it follows that the matrix ú₁ +₂₂+...+uku!
has rank k. For the cases k = 1 and k=2, explicitly establish this for
the (linearly independent) vectors ₁ = (1,0, 2), ₂ = (1,0,−1)² € R³.

Expert Solution

Step 1: Statement of the problem

According to guidelines only the first question will be solved now.

Here we have to show that the $n cross times m$ matrix $u with rightwards arrow on top v with rightwards arrow on top to the power of T$ has rank 1, where $u with rightwards arrow on top element of straight real numbers to the power of n space a n d space v with rightwards arrow on top element of straight real numbers to the power of m$ .