(i.) Suppose ū € R¹ and 7 € R™ are non-zero (column) vectors. Show that the nx m matrix uuT has rank 1. Hint: Compare columns in the matrix úv¹. (ii.) Consider the elementary basis vectors & R" (column vectors). Show that the matrix eje +e₂e₂+...+eket has rank k. Comment on the case k = n. (iii.) Suppose ū₁, ₂, ...,uk € R¹ are linearly independent. Similarly to the previous question, it follows that the matrix ₁ +₂ÚŢ +...+ūķÚT has rank k. For the cases k = 1 and k=2, explicitly establish this for the (linearly independent) vectors ₁ = (1,0, 2)¹,₂ = (1,0, −1)ª – R³.
(i.) Suppose ū € R¹ and 7 € R™ are non-zero (column) vectors. Show that the nx m matrix uuT has rank 1. Hint: Compare columns in the matrix úv¹. (ii.) Consider the elementary basis vectors & R" (column vectors). Show that the matrix eje +e₂e₂+...+eket has rank k. Comment on the case k = n. (iii.) Suppose ū₁, ₂, ...,uk € R¹ are linearly independent. Similarly to the previous question, it follows that the matrix ₁ +₂ÚŢ +...+ūķÚT has rank k. For the cases k = 1 and k=2, explicitly establish this for the (linearly independent) vectors ₁ = (1,0, 2)¹,₂ = (1,0, −1)ª – R³.
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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Transcribed Image Text:(i.) Suppose u € R" and 7 € R™ are non-zero (column) vectors. Show that
the nx m matrix uuT has rank 1.
Hint: Compare columns in the matrix úv¹.
(ii.) Consider the elementary basis vectors € R" (column vectors). Show
that the matrix eje +e₂e₂+...+eket has rank k. Comment on the
case k = n.
(iii.) Suppose ₁, ₂,...,Uk € R¹ are linearly independent. Similarly to the
previous question, it follows that the matrix ₁ +₂ÚŢ +...+ūķÚT
has rank k. For the cases k = 1 and k=2, explicitly establish this for
the (linearly independent) vectors ₁ = (1,0, 2)¹, ₂ = (1,0,−1)¹ € R³.
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