- #1
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Hi, All:
I have been tutoring linear algebra, and my student does not seem to be able
to understand a solution I proposed ( of course, I may be wrong, and/or explaining
poorly). I'm hoping someone can suggest a better explanation and/or a different solution
to this problem:
We have two vector spaces V,V' , over the same F, and we have a subspace S of V.
The goal is to construct a map T , whose kernel is precisely S. The dimensions of
S,V,W respectively work well re Rank-Nullity, i.e., DimV-DimS=DimV'. My goal is to
declare T to be zero in B_S , and then set up a bijection between the remaining basis
vectors in B_V , and the basis vectors in B_V'.
So, I propossed that:
i)We choose a basis B_S :={e_1,...,e_s} for S, extend to a basis B_V:= {e_1,e_2,...,e_b}
for V. Let B_W:={e_1',e_2',...,e_w'} ; s:=|B_S|, and so-on.
ii)Declare/define T(B_S)==0 , i.e., for each basis vector e_s in B_S, we define
T(e_s)=0
iii) Now, we set up a bijection between the basis vectors in B_V\B_S , and those in
B_W. This bijection, gives rise to an isomorphism (extending by linearity) between
Span(B_V-B_S) , and V' , so we have:
1)T(e_s)==0 , for e_s in B_S
2)T(e_s+i):=e_i'
3)T(ce_s+i+de_s+j):=cT(e_s+i)+dT(e_s+j)
Now, I'm trying to extend this to the infinite-dimensional case, but my student has
only a beginners' knowledge of set theory.
Any Suggestions?
Thanks in Advance.
P.S: She also asked me a sort-of-strange question: is there such a thing as "non-linear
algebra"? I had no idea how to answer that. anyone Know?
I have been tutoring linear algebra, and my student does not seem to be able
to understand a solution I proposed ( of course, I may be wrong, and/or explaining
poorly). I'm hoping someone can suggest a better explanation and/or a different solution
to this problem:
We have two vector spaces V,V' , over the same F, and we have a subspace S of V.
The goal is to construct a map T , whose kernel is precisely S. The dimensions of
S,V,W respectively work well re Rank-Nullity, i.e., DimV-DimS=DimV'. My goal is to
declare T to be zero in B_S , and then set up a bijection between the remaining basis
vectors in B_V , and the basis vectors in B_V'.
So, I propossed that:
i)We choose a basis B_S :={e_1,...,e_s} for S, extend to a basis B_V:= {e_1,e_2,...,e_b}
for V. Let B_W:={e_1',e_2',...,e_w'} ; s:=|B_S|, and so-on.
ii)Declare/define T(B_S)==0 , i.e., for each basis vector e_s in B_S, we define
T(e_s)=0
iii) Now, we set up a bijection between the basis vectors in B_V\B_S , and those in
B_W. This bijection, gives rise to an isomorphism (extending by linearity) between
Span(B_V-B_S) , and V' , so we have:
1)T(e_s)==0 , for e_s in B_S
2)T(e_s+i):=e_i'
3)T(ce_s+i+de_s+j):=cT(e_s+i)+dT(e_s+j)
Now, I'm trying to extend this to the infinite-dimensional case, but my student has
only a beginners' knowledge of set theory.
Any Suggestions?
Thanks in Advance.
P.S: She also asked me a sort-of-strange question: is there such a thing as "non-linear
algebra"? I had no idea how to answer that. anyone Know?