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This question examines your understanding of linear independence. Please tick all correct statements. (Try to deduce the statements from facts you know, or try to find counterexamples, i.e. find examples showing that the statements are false.) You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question will be set to zero.) ☐a. Let the vectors V1₁,...,Vm in Rº be linearly independent, and let O d. The vectors V₁,...,Vm in Rn are called linearly independent if implies a₁=a₂=...=am=0. with coefficients a=(a₁,...,am) in Rm. Then there can be coefficients B=(B₁,...,.Bm) in Rm with a B and m 0= Σ V = m =Σak k=0 V = k=1 b. The vectors x=(1,2,3) and y=(2,3,4) are linearly independent. c. If the vectors V₁,...,Vm in Rn are linearly independent, then for every j in {1,...,m} there exist a1₁,...,aj-₁,aj+1,...,am in R\{0} such that m αk Uk ΣBkUk. k=0 αk Uk m υ; = Σ j#k (*) with αk Uk. (**) a1, am ER

This question examines your understanding of linear independence. Please tick all correct statements. (Try to deduce the statements from facts you know, or try to find counterexamples, i.e. find examples showing that the statements are false.) You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question will be set to zero.) ☐a. Let the vectors V1₁,...,Vm in Rº be linearly independent, and let O d. The vectors V₁,...,Vm in Rn are called linearly independent if implies a₁=a₂=...=am=0. with coefficients a=(a₁,...,am) in Rm. Then there can be coefficients B=(B₁,...,.Bm) in Rm with a B and m 0= Σ V = m =Σak k=0 V = k=1 b. The vectors x=(1,2,3) and y=(2,3,4) are linearly independent. c. If the vectors V₁,...,Vm in Rn are linearly independent, then for every j in {1,...,m} there exist a1₁,...,aj-₁,aj+1,...,am in R\{0} such that m αk Uk ΣBkUk. k=0 αk Uk m υ; = Σ j#k (*) with αk Uk. (**) a1, am ER

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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This question examines your understanding of linear independence. Please tick all correct statements. (Try to deduce the statements from facts you know, or try to find counterexamples, i.e. find examples showing that the statements are false.) You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question will be set to zero.) a. Let the vectors V₁,...,Vm in Rn be linearly independent, and let V = with coefficients a=(a₁,...,am) in Rm. Then there can be coefficients B=(B₁1,...,ßm) in Rm with a*ß and Σβκυκ. (**) k=0 Od. The vectors V₁,...,Vm in Rh are called linearly independent if implies a₁=a₂=...=am=0. U = m m b. The vectors x=(1,2,3) and y=(2,3,4) are linearly independent. c. If the vectors V₁,...,Vm in Rº are linearly independent, then for every j in {1,...,m} there exist A₁,...,ªj-1,ªj+1,...,ªm in R\{0} such that Σακυκ (*) k=0 m ο = Σακυκ k=1 = m Σακυκ j#k with αι,..., am ΕΡ
Transcribed Image Text:This question examines your understanding of linear independence. Please tick all correct statements. (Try to deduce the statements from facts you know, or try to find counterexamples, i.e. find examples showing that the statements are false.) You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question will be set to zero.) a. Let the vectors V₁,...,Vm in Rn be linearly independent, and let V = with coefficients a=(a₁,...,am) in Rm. Then there can be coefficients B=(B₁1,...,ßm) in Rm with a*ß and Σβκυκ. (**) k=0 Od. The vectors V₁,...,Vm in Rh are called linearly independent if implies a₁=a₂=...=am=0. U = m m b. The vectors x=(1,2,3) and y=(2,3,4) are linearly independent. c. If the vectors V₁,...,Vm in Rº are linearly independent, then for every j in {1,...,m} there exist A₁,...,ªj-1,ªj+1,...,ªm in R\{0} such that Σακυκ (*) k=0 m ο = Σακυκ k=1 = m Σακυκ j#k with αι,..., am ΕΡ
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