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Linear Proofs 
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#1
Sep411, 10:33 PM

P: 36

Hello,
A(rB) = r(AB) =(rA)B where r is a real scalar and A and B are appropriately sized matrices. How to even start? A(rbij)=A(rB), but then you can't reassociate... Also, a formal proof for Tr(A^{T})=Tr(A)? It doesn't seem like enough to say the diagonal entries are unaffected by transposition.. Lastly, let A be an mxn matrix with a column consisting entirely of zeros. Show that if B is an nxp matrix, then AB has a row of zeros. I can't figure out how to make a proof of this. I know how to say what such and such entry of AB is, but I don't know how to designate an entire column. How do you formally say it will be equal to zero, then...just because the dot product of a zero vector with anything is 0? 


#2
Sep511, 08:17 AM

Math
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PF Gold
P: 39,352




#3
Sep511, 08:26 AM

P: 36

Thank you.
Also, Tr(A^{T}A)[itex]\geq[/itex]0. I can't even see how to begin... 


#4
Sep511, 08:28 AM

Math
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PF Gold
P: 39,352

Linear Proofs



#5
Sep511, 08:57 AM

Emeritus
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PF Gold
P: 9,270

All your other questions are also quite easy to answer if you just use these definitions, and the definition of the trace and the transpose: [itex]\operatorname{Tr}A=\sum_i A_{ii},\quad (A^T)_{ij}=A_{ji}[/itex]. 


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