This question examines your understanding of bases of linear vector spaces. 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. A tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if the tuple (V₁,...,Vm) is linearly independent and for every v in V, the tuple (v₁,...,V,V) is linearly dependent. b. A tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if and only if V=span(V₁...Vm). ☐ c. If a tuple (V₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V, then any tuple (W₁,...,Wn) of vectors W₁,...,W₁ in V with m>n is not a basis of V. Od. If a tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V, then any tuple (W₁,...,wn) of vectors W₁,...,n in V with m
This question examines your understanding of bases of linear vector spaces. 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. A tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if the tuple (V₁,...,Vm) is linearly independent and for every v in V, the tuple (v₁,...,V,V) is linearly dependent. b. A tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if and only if V=span(V₁...Vm). ☐ c. If a tuple (V₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V, then any tuple (W₁,...,Wn) of vectors W₁,...,W₁ in V with m>n is not a basis of V. Od. If a tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V, then any tuple (W₁,...,wn) of vectors W₁,...,n in V with m
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:This question examines your understanding of bases of linear vector spaces.
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. A tuple (V₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if the tuple (v₁,...,Vm) is linearly independent and for every v in V, the tuple (V₁,...,V,V) is
linearly dependent.
Ob. A tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if and only if V=span(v₁,...,Vm).
c.
If a tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V, then any tuple (w₁,...,wn) of vectors W₁,...,W₁ in V with m>n is not a basis of V.
Od.
If a tuple (v₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V, then any tuple (W1₁,...,wn) of vectors w₁,...,W₁ in V with m<n is not a basis of V.
e.
If a tuple (V₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V then dim(V)=m.
O f.
A tuple (V₁,...,Vm) of vectors V₁,...,Vm in V is a basis of V if and only if for every v in V, we have span(v₁,...,Vm)=span(V₁,...,V,V).
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