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Linear Algebra

Let A1,...An be m x m matrices. Suppose that B and C are m x m matrices in Span (A1,...An). Show that B-C are in span (A1,...An).
(edited 7 years ago)
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Rose86
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Anyone?
Original post by Rose86
Let A1,...An be m x m matrices. Suppose that B and C are m x m matrices in Span (A1,...An). Show that B-C are in span (A1,...An).


So, what does B is in Span (A1,...An) mean mathematically?
Similarly C.

Then what's B-C?
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Rose86
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Original post by ghostwalker
So, what does B is in Span (A1,...An) mean mathematically?
Similarly C.

Then what's B-C?

0?
Original post by Rose86
0?


Not sure what that means.
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Rose86
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Original post by ghostwalker
Not sure what that means.


I dont know Whats B-C is:frown:
Original post by Rose86
I dont know Whats B-C is:frown:


!!

Can you work out B+C ?
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Rose86
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Original post by ghostwalker
!!

Can you work out B+C ?

No😓
Reply 8
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Rose86
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Original post by Rose86
No😓


Could You?
Original post by Rose86
No😓


These should all be covered in your definitions.

B is in the Span (A1,...,An)(A_1,...,A_n) means there exists b1,..,bnb_1,..,b_n such that:

B=b1A1+...+bnAnB=b_1A_1+...+b_nA_n

Similarly for C

C=c1A1+...+cnAnC=c_1A_1+...+c_nA_n

B-C is then:

B−C=(b1A1+...+bnAn)−(c1A1+...+cnAn)B-C=(b_1A_1+...+b_nA_n)-(c_1A_1+...+c_nA_n)

Which we can rearrange to:

B−C=(b1A1−c1A1)+...+(bnAn−cnAn)B-C=(b_1A_1-c_1A_1)+...+(b_nA_n-c_nA_n)

B−C=(b1−c1)A1+...+(bn−cn)AnB-C=(b_1-c_1)A_1+...+(b_n-c_n)A_n

b1−c1b_1-c_1 are just numbers, so this shows that B-C is in the Span (A1,...,An)(A_1,...,A_n)

Note: I have avoided the use of terms such as "vector space", "field", etc. as you've not mentioned them, but it is the axioms of those structures that you're using in each step of the proof.
Reply 10
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Rose86
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Original post by ghostwalker
These should all be covered in your definitions.

B is in the Span (A1,...,An)(A_1,...,A_n) means there exists b1,..,bnb_1,..,b_n such that:

B=b1A1+...+bnAnB=b_1A_1+...+b_nA_n

Similarly for C

C=c1A1+...+cnAnC=c_1A_1+...+c_nA_n

B-C is then:

B−C=(b1A1+...+bnAn)−(c1A1+...+cnAn)B-C=(b_1A_1+...+b_nA_n)-(c_1A_1+...+c_nA_n)

Which we can rearrange to:

B−C=(b1A1−c1A1)+...+(bnAn−cnAn)B-C=(b_1A_1-c_1A_1)+...+(b_nA_n-c_nA_n)

B−C=(b1−c1)A1+...+(bn−cn)AnB-C=(b_1-c_1)A_1+...+(b_n-c_n)A_n

b1−c1b_1-c_1 are just numbers, so this shows that B-C is in the Span (A1,...,An)(A_1,...,A_n)

Note: I have avoided the use of terms such as "vector space", "field", etc. as you've not mentioned them, but it is the axioms of those structures that you're using in each step of the proof.


Thank You!

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