(ii.) Consider the elementary basis vectors ; & R" (column vectors). ShowTthat the matrix eje +e₂e +...+eker has rank k. Comment on thecase k = n.

Given that ei→∈Rn are elementary basis.We need to show that e1→e1→T+e2→e2→T+…+ek→ek→T has rank k.We…

Answered: Consider the elementary basis vectors &…

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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Question

(ii.) Consider the elementary basis vectors ; & R" (column vectors). Show
T
that the matrix eje +e₂e +...+eker has rank k. Comment on the
case k = n.

Expert Solution

Step 1: Introduction

Given that $\vec{e_{i}} \in R^{n}$ are elementary basis.

We need to show that $\vec{e_{1}} {\vec{e_{1}}}^{T} + \vec{e_{2}} {\vec{e_{2}}}^{T} + \dots + \vec{e_{k}} {\vec{e_{k}}}^{T}$ has rank $k$ .

We know that $\vec{e_{i}}$ is a column vector of size $n \times 1$ , whose $i^{th}$ row, column $1$ entry is $1$ and all other entries are $0$ .

That is $stack e subscript i with rightwards arrow on top equals open square brackets table row 0 row vertical ellipsis row 0 row cell 1 space left parenthesis i to the power of t h end exponent space text row end text right parenthesis end cell row 0 row vertical ellipsis row 0 end table close square brackets$ .