Show that exists a subspace $U$ so that the orthogonal projection of $v$ onto it is of any specific length $alpha < ||v||$

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Given the inner product space $V$ such that $dim V > 1$



Show that for every $alpha in mathbb R$ and $v in V$ such that $0le alpha le || v||$ exists a subspace U of V such that the orthogonal projection of $v$ onto $U$ is of length $alpha$.




I think this has something to do with representing U in an orthonormal basis $u_1$ for example and then saying $P_U(v)=<v,u_1>v$. But I'm not really sure about this.




linear-algebra projection




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This question has an open bounty worth +50
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  • 3




    Let $w$ be a vector perpendicular to $v$. When you project $v$ onto $textspanw$, the length is 0. When you project $v$ onto $textspanv$, the length is $|v|$. What happens when you project onto $textspanu$, where $u$ is on the line segment connecting $v$ to $w$?
    – Mike Earnest
    2 days ago










  • @mike I think I see that if I make the line segmant u longer than the length of the projection will get shorter?
    – Jason
    2 days ago














up vote
0
down vote

favorite













Given the inner product space $V$ such that $dim V > 1$



Show that for every $alpha in mathbb R$ and $v in V$ such that $0le alpha le || v||$ exists a subspace U of V such that the orthogonal projection of $v$ onto $U$ is of length $alpha$.




I think this has something to do with representing U in an orthonormal basis $u_1$ for example and then saying $P_U(v)=<v,u_1>v$. But I'm not really sure about this.




linear-algebra projection




share|cite|improve this question













This question has an open bounty worth +50
reputation from Jason ending ending at 2018-08-13 16:05:31Z">in 7 days.


The current answers do not contain enough detail.











  • 3




    Let $w$ be a vector perpendicular to $v$. When you project $v$ onto $textspanw$, the length is 0. When you project $v$ onto $textspanv$, the length is $|v|$. What happens when you project onto $textspanu$, where $u$ is on the line segment connecting $v$ to $w$?
    – Mike Earnest
    2 days ago










  • @mike I think I see that if I make the line segmant u longer than the length of the projection will get shorter?
    – Jason
    2 days ago












up vote
0
down vote

favorite









up vote
0
down vote

favorite












Given the inner product space $V$ such that $dim V > 1$



Show that for every $alpha in mathbb R$ and $v in V$ such that $0le alpha le || v||$ exists a subspace U of V such that the orthogonal projection of $v$ onto $U$ is of length $alpha$.




I think this has something to do with representing U in an orthonormal basis $u_1$ for example and then saying $P_U(v)=<v,u_1>v$. But I'm not really sure about this.




linear-algebra projection




share|cite|improve this question












Given the inner product space $V$ such that $dim V > 1$



Show that for every $alpha in mathbb R$ and $v in V$ such that $0le alpha le || v||$ exists a subspace U of V such that the orthogonal projection of $v$ onto $U$ is of length $alpha$.




I think this has something to do with representing U in an orthonormal basis $u_1$ for example and then saying $P_U(v)=<v,u_1>v$. But I'm not really sure about this.





linear-algebra projection





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked 2 days ago









Jason

1,09111327




1,09111327






This question has an open bounty worth +50
reputation from Jason ending ending at 2018-08-13 16:05:31Z">in 7 days.


The current answers do not contain enough detail.








This question has an open bounty worth +50
reputation from Jason ending ending at 2018-08-13 16:05:31Z">in 7 days.


The current answers do not contain enough detail.









  • 3




    Let $w$ be a vector perpendicular to $v$. When you project $v$ onto $textspanw$, the length is 0. When you project $v$ onto $textspanv$, the length is $|v|$. What happens when you project onto $textspanu$, where $u$ is on the line segment connecting $v$ to $w$?
    – Mike Earnest
    2 days ago










  • @mike I think I see that if I make the line segmant u longer than the length of the projection will get shorter?
    – Jason
    2 days ago












  • 3




    Let $w$ be a vector perpendicular to $v$. When you project $v$ onto $textspanw$, the length is 0. When you project $v$ onto $textspanv$, the length is $|v|$. What happens when you project onto $textspanu$, where $u$ is on the line segment connecting $v$ to $w$?
    – Mike Earnest
    2 days ago










  • @mike I think I see that if I make the line segmant u longer than the length of the projection will get shorter?
    – Jason
    2 days ago







3




3




Let $w$ be a vector perpendicular to $v$. When you project $v$ onto $textspanw$, the length is 0. When you project $v$ onto $textspanv$, the length is $|v|$. What happens when you project onto $textspanu$, where $u$ is on the line segment connecting $v$ to $w$?
– Mike Earnest
2 days ago




Let $w$ be a vector perpendicular to $v$. When you project $v$ onto $textspanw$, the length is 0. When you project $v$ onto $textspanv$, the length is $|v|$. What happens when you project onto $textspanu$, where $u$ is on the line segment connecting $v$ to $w$?
– Mike Earnest
2 days ago












@mike I think I see that if I make the line segmant u longer than the length of the projection will get shorter?
– Jason
2 days ago




@mike I think I see that if I make the line segmant u longer than the length of the projection will get shorter?
– Jason
2 days ago










1 Answer
1






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1
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Hint: Visualize this in $mathbb R^2$, projecting onto one-dimensional subspaces. The proof will generalize.






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  • I think I can visualize it but I'm having troible writing down a proof
    – Jason
    2 days ago










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













Hint: Visualize this in $mathbb R^2$, projecting onto one-dimensional subspaces. The proof will generalize.






share|cite|improve this answer





















  • I think I can visualize it but I'm having troible writing down a proof
    – Jason
    2 days ago














up vote
1
down vote













Hint: Visualize this in $mathbb R^2$, projecting onto one-dimensional subspaces. The proof will generalize.






share|cite|improve this answer





















  • I think I can visualize it but I'm having troible writing down a proof
    – Jason
    2 days ago












up vote
1
down vote










up vote
1
down vote









Hint: Visualize this in $mathbb R^2$, projecting onto one-dimensional subspaces. The proof will generalize.






share|cite|improve this answer













Hint: Visualize this in $mathbb R^2$, projecting onto one-dimensional subspaces. The proof will generalize.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered 2 days ago









Zarrax

34.6k248102




34.6k248102











  • I think I can visualize it but I'm having troible writing down a proof
    – Jason
    2 days ago
















  • I think I can visualize it but I'm having troible writing down a proof
    – Jason
    2 days ago















I think I can visualize it but I'm having troible writing down a proof
– Jason
2 days ago




I think I can visualize it but I'm having troible writing down a proof
– Jason
2 days ago












 

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