Linear Algebra, Least squares

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I'm going through the Linear Algebra course on Brilliant.org, but can't figure out how I should know that $frac173$ is the best value for $c$ to fill in here.

My two questions are:

1) Why do we want to minimize the distance of

$$3c^2-34c+101$$

2) How can you derive where this distance is minimised?

edited Jul 15 at 11:10

mechanodroid

22.3k52041

asked Jul 15 at 10:46

Daphne

112

add a commentÂ |Â

up vote
1
down vote

favorite

I'm going through the Linear Algebra course on Brilliant.org, but can't figure out how I should know that $frac173$ is the best value for $c$ to fill in here.

My two questions are:

1) Why do we want to minimize the distance of

$$3c^2-34c+101$$

2) How can you derive where this distance is minimised?

edited Jul 15 at 11:10

mechanodroid

22.3k52041

asked Jul 15 at 10:46

Daphne

112

add a commentÂ |Â

up vote
1
down vote

favorite

I'm going through the Linear Algebra course on Brilliant.org, but can't figure out how I should know that $frac173$ is the best value for $c$ to fill in here.

My two questions are:

1) Why do we want to minimize the distance of

$$3c^2-34c+101$$

2) How can you derive where this distance is minimised?

edited Jul 15 at 11:10

mechanodroid

22.3k52041

asked Jul 15 at 10:46

Daphne

112

I'm going through the Linear Algebra course on Brilliant.org, but can't figure out how I should know that $frac173$ is the best value for $c$ to fill in here.

My two questions are:

1) Why do we want to minimize the distance of

$$3c^2-34c+101$$

2) How can you derive where this distance is minimised?

edited Jul 15 at 11:10

mechanodroid

22.3k52041

asked Jul 15 at 10:46

Daphne

112

edited Jul 15 at 11:10

mechanodroid

22.3k52041

edited Jul 15 at 11:10

mechanodroid

22.3k52041

edited Jul 15 at 11:10

mechanodroid

22.3k52041

asked Jul 15 at 10:46

Daphne

112

asked Jul 15 at 10:46

Daphne

112

asked Jul 15 at 10:46

Daphne

112

add a commentÂ |Â

2 Answers
2

active

oldest

votes

up vote
1
down vote

accepted

You wish to minimize the distance of $cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3$ to the vector $pmatrix5 \ 0 \ 0$.

We have

$$left|cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3 - pmatrix5 \ 0 \ 0right|^2 = left|pmatrixc-7 \ c-4 \ c-6right|^2 = (c-7)^2+(c-4)^2+(c-6)^2 = 3c^2-34c+101$$

To minimize the polynomial $3c^2-34c+101$, write it as

$$3c^2-34c+101 = 3underbraceleft(c - frac173right)^2_ge 0 + frac143$$

so the minimum is attained for $c = frac173$ and it is equal to $frac143$.

answered Jul 15 at 11:07

mechanodroid

22.3k52041

Thanks! So then this is the closest we can get to a real answer right? The "best" solution we can derive given there is no real possible solution
â€“Â Daphne
Jul 15 at 11:22

@Daphne Yes, the closest solution of the form $pmatrixc \ -2$ is $pmatrixfrac173 \ -2$.
â€“Â mechanodroid
Jul 15 at 11:39

add a commentÂ |Â

up vote
0
down vote

Find the derivative of the polynomial given. The root is the number 17/3...

answered Jul 15 at 10:52

dmtri

347213

So you mean, first remove the ^2 from the polynomial so you are left with 3c - 34c + 101. Then take the derivative so you get 17/3 correct?
â€“Â Daphne
Jul 15 at 11:04

Can you give some intuition as in why we take the derivative in this case?
â€“Â Daphne
Jul 15 at 11:04

6c-34.please note that minimum of the poly is the root of the last you mentioned
â€“Â dmtri
Jul 15 at 11:06

You need to find the minimum of the function f=3c^2-34+101. One way to do this is with the help of derivative...
â€“Â dmtri
Jul 15 at 11:09

add a commentÂ |Â

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

up vote
1
down vote

accepted

You wish to minimize the distance of $cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3$ to the vector $pmatrix5 \ 0 \ 0$.

We have

$$left|cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3 - pmatrix5 \ 0 \ 0right|^2 = left|pmatrixc-7 \ c-4 \ c-6right|^2 = (c-7)^2+(c-4)^2+(c-6)^2 = 3c^2-34c+101$$

To minimize the polynomial $3c^2-34c+101$, write it as

$$3c^2-34c+101 = 3underbraceleft(c - frac173right)^2_ge 0 + frac143$$

so the minimum is attained for $c = frac173$ and it is equal to $frac143$.

answered Jul 15 at 11:07

mechanodroid

22.3k52041

Thanks! So then this is the closest we can get to a real answer right? The "best" solution we can derive given there is no real possible solution
â€“Â Daphne
Jul 15 at 11:22

@Daphne Yes, the closest solution of the form $pmatrixc \ -2$ is $pmatrixfrac173 \ -2$.
â€“Â mechanodroid
Jul 15 at 11:39

add a commentÂ |Â

up vote
1
down vote

accepted

You wish to minimize the distance of $cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3$ to the vector $pmatrix5 \ 0 \ 0$.

We have

$$left|cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3 - pmatrix5 \ 0 \ 0right|^2 = left|pmatrixc-7 \ c-4 \ c-6right|^2 = (c-7)^2+(c-4)^2+(c-6)^2 = 3c^2-34c+101$$

To minimize the polynomial $3c^2-34c+101$, write it as

$$3c^2-34c+101 = 3underbraceleft(c - frac173right)^2_ge 0 + frac143$$

so the minimum is attained for $c = frac173$ and it is equal to $frac143$.

answered Jul 15 at 11:07

mechanodroid

22.3k52041

Thanks! So then this is the closest we can get to a real answer right? The "best" solution we can derive given there is no real possible solution
â€“Â Daphne
Jul 15 at 11:22

@Daphne Yes, the closest solution of the form $pmatrixc \ -2$ is $pmatrixfrac173 \ -2$.
â€“Â mechanodroid
Jul 15 at 11:39

add a commentÂ |Â

up vote
1
down vote

accepted

You wish to minimize the distance of $cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3$ to the vector $pmatrix5 \ 0 \ 0$.

We have

$$left|cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3 - pmatrix5 \ 0 \ 0right|^2 = left|pmatrixc-7 \ c-4 \ c-6right|^2 = (c-7)^2+(c-4)^2+(c-6)^2 = 3c^2-34c+101$$

To minimize the polynomial $3c^2-34c+101$, write it as

$$3c^2-34c+101 = 3underbraceleft(c - frac173right)^2_ge 0 + frac143$$

so the minimum is attained for $c = frac173$ and it is equal to $frac143$.

answered Jul 15 at 11:07

mechanodroid

22.3k52041

You wish to minimize the distance of $cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3$ to the vector $pmatrix5 \ 0 \ 0$.

We have

$$left|cpmatrix1 \ 1 \ 1 -2 pmatrix1 \ 2 \ 3 - pmatrix5 \ 0 \ 0right|^2 = left|pmatrixc-7 \ c-4 \ c-6right|^2 = (c-7)^2+(c-4)^2+(c-6)^2 = 3c^2-34c+101$$

To minimize the polynomial $3c^2-34c+101$, write it as

$$3c^2-34c+101 = 3underbraceleft(c - frac173right)^2_ge 0 + frac143$$

so the minimum is attained for $c = frac173$ and it is equal to $frac143$.

answered Jul 15 at 11:07

mechanodroid

22.3k52041

answered Jul 15 at 11:07

mechanodroid

22.3k52041

answered Jul 15 at 11:07

mechanodroid

22.3k52041

answered Jul 15 at 11:07

mechanodroid

22.3k52041

Thanks! So then this is the closest we can get to a real answer right? The "best" solution we can derive given there is no real possible solution
â€“Â Daphne
Jul 15 at 11:22

@Daphne Yes, the closest solution of the form $pmatrixc \ -2$ is $pmatrixfrac173 \ -2$.
â€“Â mechanodroid
Jul 15 at 11:39

add a commentÂ |Â

Thanks! So then this is the closest we can get to a real answer right? The "best" solution we can derive given there is no real possible solution
â€“Â Daphne
Jul 15 at 11:22

@Daphne Yes, the closest solution of the form $pmatrixc \ -2$ is $pmatrixfrac173 \ -2$.
â€“Â mechanodroid
Jul 15 at 11:39

Thanks! So then this is the closest we can get to a real answer right? The "best" solution we can derive given there is no real possible solution
â€“Â Daphne
Jul 15 at 11:22

@Daphne Yes, the closest solution of the form $pmatrixc \ -2$ is $pmatrixfrac173 \ -2$.
â€“Â mechanodroid
Jul 15 at 11:39

add a commentÂ |Â

up vote
0
down vote

Find the derivative of the polynomial given. The root is the number 17/3...

answered Jul 15 at 10:52

dmtri

347213

So you mean, first remove the ^2 from the polynomial so you are left with 3c - 34c + 101. Then take the derivative so you get 17/3 correct?
â€“Â Daphne
Jul 15 at 11:04

Can you give some intuition as in why we take the derivative in this case?
â€“Â Daphne
Jul 15 at 11:04

6c-34.please note that minimum of the poly is the root of the last you mentioned
â€“Â dmtri
Jul 15 at 11:06

You need to find the minimum of the function f=3c^2-34+101. One way to do this is with the help of derivative...
â€“Â dmtri
Jul 15 at 11:09

add a commentÂ |Â

up vote
0
down vote

Find the derivative of the polynomial given. The root is the number 17/3...

answered Jul 15 at 10:52

dmtri

347213

So you mean, first remove the ^2 from the polynomial so you are left with 3c - 34c + 101. Then take the derivative so you get 17/3 correct?
â€“Â Daphne
Jul 15 at 11:04

Can you give some intuition as in why we take the derivative in this case?
â€“Â Daphne
Jul 15 at 11:04

6c-34.please note that minimum of the poly is the root of the last you mentioned
â€“Â dmtri
Jul 15 at 11:06

You need to find the minimum of the function f=3c^2-34+101. One way to do this is with the help of derivative...
â€“Â dmtri
Jul 15 at 11:09

add a commentÂ |Â

up vote
0
down vote

Find the derivative of the polynomial given. The root is the number 17/3...

answered Jul 15 at 10:52

dmtri

347213

Find the derivative of the polynomial given. The root is the number 17/3...

answered Jul 15 at 10:52

dmtri

347213

answered Jul 15 at 10:52

dmtri

347213

answered Jul 15 at 10:52

dmtri

347213

answered Jul 15 at 10:52

dmtri

347213

So you mean, first remove the ^2 from the polynomial so you are left with 3c - 34c + 101. Then take the derivative so you get 17/3 correct?
â€“Â Daphne
Jul 15 at 11:04

Can you give some intuition as in why we take the derivative in this case?
â€“Â Daphne
Jul 15 at 11:04

6c-34.please note that minimum of the poly is the root of the last you mentioned
â€“Â dmtri
Jul 15 at 11:06

You need to find the minimum of the function f=3c^2-34+101. One way to do this is with the help of derivative...
â€“Â dmtri
Jul 15 at 11:09

add a commentÂ |Â

So you mean, first remove the ^2 from the polynomial so you are left with 3c - 34c + 101. Then take the derivative so you get 17/3 correct?
â€“Â Daphne
Jul 15 at 11:04

Can you give some intuition as in why we take the derivative in this case?
â€“Â Daphne
Jul 15 at 11:04

6c-34.please note that minimum of the poly is the root of the last you mentioned
â€“Â dmtri
Jul 15 at 11:06

You need to find the minimum of the function f=3c^2-34+101. One way to do this is with the help of derivative...
â€“Â dmtri
Jul 15 at 11:09

So you mean, first remove the ^2 from the polynomial so you are left with 3c - 34c + 101. Then take the derivative so you get 17/3 correct?
â€“Â Daphne
Jul 15 at 11:04

Can you give some intuition as in why we take the derivative in this case?
â€“Â Daphne
Jul 15 at 11:04

6c-34.please note that minimum of the poly is the root of the last you mentioned
â€“Â dmtri
Jul 15 at 11:06

You need to find the minimum of the function f=3c^2-34+101. One way to do this is with the help of derivative...
â€“Â dmtri
Jul 15 at 11:09

add a commentÂ |Â

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