Kernel to Orthogonal Matrix relationship

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Let $B$ be a matrix. How can I tell which ($mathrmimg(B)$, $mathrmker(B^T)$, $mathrmimg(B^T)$) spaces are necessarily orthogonal to $mathrmker(B)$ under standard dot product?



What are the implications of orthogonality from dot product?




linear-algebra matrices linear-transformations orthogonality




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  • It's not clear what you mean by the second part of your question, "what are the implications of orthogonality from dot product?"
    – Omnomnomnom
    Aug 3 at 13:50










  • @Omnomnomnom how do we derive orthogonality from dot product procedures? Sorry for the lack of clarity.
    – seekingalpha23
    Aug 3 at 13:51










  • Are $A$ and $B$ supposed to be different matrices? Or, are you asking why $ker(A)$ is orthogonal to $operatornameimage(A^T)$?
    – Omnomnomnom
    Aug 3 at 13:53











  • @Omnomnomnom sorry I edited as suggested
    – seekingalpha23
    Aug 3 at 14:13














up vote
1
down vote

favorite












Let $B$ be a matrix. How can I tell which ($mathrmimg(B)$, $mathrmker(B^T)$, $mathrmimg(B^T)$) spaces are necessarily orthogonal to $mathrmker(B)$ under standard dot product?



What are the implications of orthogonality from dot product?




linear-algebra matrices linear-transformations orthogonality




share|cite|improve this question





















  • It's not clear what you mean by the second part of your question, "what are the implications of orthogonality from dot product?"
    – Omnomnomnom
    Aug 3 at 13:50










  • @Omnomnomnom how do we derive orthogonality from dot product procedures? Sorry for the lack of clarity.
    – seekingalpha23
    Aug 3 at 13:51










  • Are $A$ and $B$ supposed to be different matrices? Or, are you asking why $ker(A)$ is orthogonal to $operatornameimage(A^T)$?
    – Omnomnomnom
    Aug 3 at 13:53











  • @Omnomnomnom sorry I edited as suggested
    – seekingalpha23
    Aug 3 at 14:13












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let $B$ be a matrix. How can I tell which ($mathrmimg(B)$, $mathrmker(B^T)$, $mathrmimg(B^T)$) spaces are necessarily orthogonal to $mathrmker(B)$ under standard dot product?



What are the implications of orthogonality from dot product?




linear-algebra matrices linear-transformations orthogonality




share|cite|improve this question













Let $B$ be a matrix. How can I tell which ($mathrmimg(B)$, $mathrmker(B^T)$, $mathrmimg(B^T)$) spaces are necessarily orthogonal to $mathrmker(B)$ under standard dot product?



What are the implications of orthogonality from dot product?





linear-algebra matrices linear-transformations orthogonality





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 3 at 13:59
























asked Aug 3 at 13:44









seekingalpha23

156




156











  • It's not clear what you mean by the second part of your question, "what are the implications of orthogonality from dot product?"
    – Omnomnomnom
    Aug 3 at 13:50










  • @Omnomnomnom how do we derive orthogonality from dot product procedures? Sorry for the lack of clarity.
    – seekingalpha23
    Aug 3 at 13:51










  • Are $A$ and $B$ supposed to be different matrices? Or, are you asking why $ker(A)$ is orthogonal to $operatornameimage(A^T)$?
    – Omnomnomnom
    Aug 3 at 13:53











  • @Omnomnomnom sorry I edited as suggested
    – seekingalpha23
    Aug 3 at 14:13
















  • It's not clear what you mean by the second part of your question, "what are the implications of orthogonality from dot product?"
    – Omnomnomnom
    Aug 3 at 13:50










  • @Omnomnomnom how do we derive orthogonality from dot product procedures? Sorry for the lack of clarity.
    – seekingalpha23
    Aug 3 at 13:51










  • Are $A$ and $B$ supposed to be different matrices? Or, are you asking why $ker(A)$ is orthogonal to $operatornameimage(A^T)$?
    – Omnomnomnom
    Aug 3 at 13:53











  • @Omnomnomnom sorry I edited as suggested
    – seekingalpha23
    Aug 3 at 14:13















It's not clear what you mean by the second part of your question, "what are the implications of orthogonality from dot product?"
– Omnomnomnom
Aug 3 at 13:50




It's not clear what you mean by the second part of your question, "what are the implications of orthogonality from dot product?"
– Omnomnomnom
Aug 3 at 13:50












@Omnomnomnom how do we derive orthogonality from dot product procedures? Sorry for the lack of clarity.
– seekingalpha23
Aug 3 at 13:51




@Omnomnomnom how do we derive orthogonality from dot product procedures? Sorry for the lack of clarity.
– seekingalpha23
Aug 3 at 13:51












Are $A$ and $B$ supposed to be different matrices? Or, are you asking why $ker(A)$ is orthogonal to $operatornameimage(A^T)$?
– Omnomnomnom
Aug 3 at 13:53





Are $A$ and $B$ supposed to be different matrices? Or, are you asking why $ker(A)$ is orthogonal to $operatornameimage(A^T)$?
– Omnomnomnom
Aug 3 at 13:53













@Omnomnomnom sorry I edited as suggested
– seekingalpha23
Aug 3 at 14:13




@Omnomnomnom sorry I edited as suggested
– seekingalpha23
Aug 3 at 14:13















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