Q4. Let vi,..., vk E R". Assume that Span{vỉ, ..., v} = R". In this problem you will show thatn< k, i.e., R" cannot be spanned by fewer than n vectors (as you might intuitively expect).v11V21Uk1V12V22Vk2Show that the system of equationsUk =(a) Let viU2 =V2nVknV1nV11x1+ V21X2+ · · ·+ Uk1Xk = b1V12X1 + v22x2 ++ Vk2Xk =b2Vinx1 + v2n X2 ++ vknXk =bnis consistent for all b1, b2,..., bn E R.(b) Using part (a), show that n < k.

Solution (a): Given that v1→=v11v12⋮v1n, v2→=v21v22⋮v2n, …, vk→=vk1vk2⋮vkn . . .…

Q4. Let vi,..., vE E R". Assume that Span{vi,..., T} = R". In this problem you will show that n < k, i.e., R" cannot be spanned by fewer than n vectors (as you might intuitively expect). V11 V21 Uk1 V12 V22 Uk2 Show that the system of equations (a) Let vj = Vin V2n Ukn Vi1x1+ v21X2+ · · ·+ vk1Xk = b1 + vk2Xk = b2 V12X1+ V22X2 + • · · VinX1+ V2n2 + · ··+ VknXk = bn is consistent for all b1, b2, ..., bn E R. (b) Using part (a), show that n< k.

Q4. Let vi,..., vE E R". Assume that Span{vi,..., T} = R". In this problem you will show that n < k, i.e., R" cannot be spanned by fewer than n vectors (as you might intuitively expect). V11 V21 Uk1 V12 V22 Uk2 Show that the system of equations (a) Let vj = Vin V2n Ukn Vi1x1+ v21X2+ · · ·+ vk1Xk = b1 + vk2Xk = b2 V12X1+ V22X2 + • · · VinX1+ V2n2 + · ··+ VknXk = bn is consistent for all b1, b2, ..., bn E R. (b) Using part (a), show that n< k.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.3: Spanning Sets And Linear Independence

Problem 21EQ

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