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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.

Advanced Engineering Mathematics
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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Q4. Let vi,..., vk E R". Assume that Span{vỉ, ..., v} = 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 Vk2 Show that the system of equations Uk = (a) Let vi U2 = V2n Vkn V1n V11x1+ V21X2+ · · · + Uk1Xk = b1 V12X1 + v22x2 + + Vk2Xk = b2 Vinx1 + v2n X2 + + vknXk = bn is consistent for all b1, b2,..., bn E R. (b) Using part (a), show that n < k.
Transcribed Image Text:Q4. Let vi,..., vk E R". Assume that Span{vỉ, ..., v} = 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 Vk2 Show that the system of equations Uk = (a) Let vi U2 = V2n Vkn V1n V11x1+ V21X2+ · · · + Uk1Xk = b1 V12X1 + v22x2 + + Vk2Xk = b2 Vinx1 + v2n X2 + + vknXk = bn is consistent for all b1, b2,..., bn E R. (b) Using part (a), show that n < k.
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