- #1

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i just don't know how to find the rank(A+B) from the column vectors of (A+B) ... or there's another way to prove?

thanks a lot

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- Thread starter boombaby
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- #1

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i just don't know how to find the rank(A+B) from the column vectors of (A+B) ... or there's another way to prove?

thanks a lot

- #2

matt grime

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- #3

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without your help i might still be stuck with finding out which a_i+b_i is the base vector of A+B...

- #4

matt grime

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But, anything in the span of <a_i+b_i> is in the span of <a_i,b_j>, which gives the answer.

Also, the number of columns of A is not its rank - you don't just take the columns a_1,..,a_rank(A)

- #5

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but i think... the rank of <a_i,b_j> dose not equal to rank(A)+ rank(B). if A=B ,then <a_i,b_j>=<a_i> .am i right?

so can i say rank(A+B)<= rank<A,B> <=rank A+rank B ? thanks for help

- #6

matt grime

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You are on the right lines - think about how to write things clearly.

- #7

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there is nothing that says r of them will form a basis.

i'm not quite clear about this.

rank(A) is defined (in my book) to be dim<a_1 ,a_2,a_n>. and so definitely r of them will form a basis of the span <a_1,a_2,...a_n> ,so why can't i make some choice of a_i to be the basis? sorry for asking if i misunderstand something.....but i just want to make sure of it.

thanks again

- #8

matt grime

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I was being dense - this is just a replacement lemma.

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