Show that mapping from polynomial to a real number is a linear mapping.
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Denote by $$Pi_2left([0,1]right) = t mapsto c_0 + c_1t + c_2t^2 : c_k in mathbbR textand tin [0,1] $$
the set of polynomials of at most degree 2 with real coefficients defined over $[0,1]$. Equipped with pointwise addition and multiplication by scarlars this set and operations defines a linear space over $mathbb R$.
- show that $Pi_2left([0,1]right) oversetTto mathbb R$ defined by $p mapsto T(p) = p(1) - p(0)$ is a linear mapping.
To show its a linear mapping we need to prove: $$1. f(x+y) = f(x) + f(y)$$ $$2. f(lambda(x)) = lambda(x).$$
$1.$ Let $p,q$ be elements of the 2-degree polynomial from the question such that $p(t) = p_o + p_1x + p_2x^2 $and $q(t) = q_o + q_1x + q_2x^2.$
$T(p+q) = T([p_o + p_1x + p_2x^2] + [q_o + q_1x + q_2x^2]) = (p(1) + q(1)) - (p(0)+ q(0)) =[p(1) -p(0)] + [q(1)-q(0)] = T(p) + T(q)$
$2$ Let $lambda$ be a scalar that is an element of the set of real numbers and p be an element of the 2-degree polynomial such that $ p(t) = p_o + p_1x + p_2x^2.$
$T(λp) = T(λp_o + λp_1x + p_2x^2) = λp(1) - λp(0) = λ(p(1) - p(0)) = λT(p) $ as required.
Would this be correct?
linear-algebra functions
edited Aug 28 at 14:16
amWhy
190k26221433
asked Aug 28 at 13:43
Randy Rogers
726
 |Â
show 4 more comments
up vote
0
down vote
favorite
Denote by $$Pi_2left([0,1]right) = t mapsto c_0 + c_1t + c_2t^2 : c_k in mathbbR textand tin [0,1] $$
the set of polynomials of at most degree 2 with real coefficients defined over $[0,1]$. Equipped with pointwise addition and multiplication by scarlars this set and operations defines a linear space over $mathbb R$.
- show that $Pi_2left([0,1]right) oversetTto mathbb R$ defined by $p mapsto T(p) = p(1) - p(0)$ is a linear mapping.
To show its a linear mapping we need to prove: $$1. f(x+y) = f(x) + f(y)$$ $$2. f(lambda(x)) = lambda(x).$$
$1.$ Let $p,q$ be elements of the 2-degree polynomial from the question such that $p(t) = p_o + p_1x + p_2x^2 $and $q(t) = q_o + q_1x + q_2x^2.$
$T(p+q) = T([p_o + p_1x + p_2x^2] + [q_o + q_1x + q_2x^2]) = (p(1) + q(1)) - (p(0)+ q(0)) =[p(1) -p(0)] + [q(1)-q(0)] = T(p) + T(q)$
$2$ Let $lambda$ be a scalar that is an element of the set of real numbers and p be an element of the 2-degree polynomial such that $ p(t) = p_o + p_1x + p_2x^2.$
$T(λp) = T(λp_o + λp_1x + p_2x^2) = λp(1) - λp(0) = λ(p(1) - p(0)) = λT(p) $ as required.
Would this be correct?
linear-algebra functions
edited Aug 28 at 14:16
amWhy
190k26221433
asked Aug 28 at 13:43
Randy Rogers
726
1
It's fine, but you don't need to write down the express form of each polynomial.
– DonAntonio
Aug 28 at 13:46
welcome to MSE. please try avoid using in embed pictures, but instead try using MathJax
– Jneven
Aug 28 at 14:16
Sure, but whats wrong with using embed pictures? It saves me time and i have alot of studying to do @Jneven
– Randy Rogers
Aug 28 at 14:18
other users won't be able to find the question if they are looking for it. this is a community. asking questions is a big part of it. but the information should be as available as possible.
– Jneven
Aug 28 at 14:20
1
Randy, this site is not for you, it's for everyone. Using text and MathJax instead of pictures improves the overall interface of the site, the readability for people who use screen readers, and search results both on the site and from Google when people are looking for related topics and questions. Remember that the questions you ask are likely to be similar to questions that other people have elsewhere. Not everything is confined around your exams.
– Asaf Karagila♦
Aug 28 at 15:19
 |Â
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Denote by $$Pi_2left([0,1]right) = t mapsto c_0 + c_1t + c_2t^2 : c_k in mathbbR textand tin [0,1] $$
the set of polynomials of at most degree 2 with real coefficients defined over $[0,1]$. Equipped with pointwise addition and multiplication by scarlars this set and operations defines a linear space over $mathbb R$.
- show that $Pi_2left([0,1]right) oversetTto mathbb R$ defined by $p mapsto T(p) = p(1) - p(0)$ is a linear mapping.
To show its a linear mapping we need to prove: $$1. f(x+y) = f(x) + f(y)$$ $$2. f(lambda(x)) = lambda(x).$$
$1.$ Let $p,q$ be elements of the 2-degree polynomial from the question such that $p(t) = p_o + p_1x + p_2x^2 $and $q(t) = q_o + q_1x + q_2x^2.$
$T(p+q) = T([p_o + p_1x + p_2x^2] + [q_o + q_1x + q_2x^2]) = (p(1) + q(1)) - (p(0)+ q(0)) =[p(1) -p(0)] + [q(1)-q(0)] = T(p) + T(q)$
$2$ Let $lambda$ be a scalar that is an element of the set of real numbers and p be an element of the 2-degree polynomial such that $ p(t) = p_o + p_1x + p_2x^2.$
$T(λp) = T(λp_o + λp_1x + p_2x^2) = λp(1) - λp(0) = λ(p(1) - p(0)) = λT(p) $ as required.
Would this be correct?
linear-algebra functions
edited Aug 28 at 14:16
amWhy
190k26221433
asked Aug 28 at 13:43
Randy Rogers
726
Denote by $$Pi_2left([0,1]right) = t mapsto c_0 + c_1t + c_2t^2 : c_k in mathbbR textand tin [0,1] $$
the set of polynomials of at most degree 2 with real coefficients defined over $[0,1]$. Equipped with pointwise addition and multiplication by scarlars this set and operations defines a linear space over $mathbb R$.
- show that $Pi_2left([0,1]right) oversetTto mathbb R$ defined by $p mapsto T(p) = p(1) - p(0)$ is a linear mapping.
To show its a linear mapping we need to prove: $$1. f(x+y) = f(x) + f(y)$$ $$2. f(lambda(x)) = lambda(x).$$
$1.$ Let $p,q$ be elements of the 2-degree polynomial from the question such that $p(t) = p_o + p_1x + p_2x^2 $and $q(t) = q_o + q_1x + q_2x^2.$
$T(p+q) = T([p_o + p_1x + p_2x^2] + [q_o + q_1x + q_2x^2]) = (p(1) + q(1)) - (p(0)+ q(0)) =[p(1) -p(0)] + [q(1)-q(0)] = T(p) + T(q)$
$2$ Let $lambda$ be a scalar that is an element of the set of real numbers and p be an element of the 2-degree polynomial such that $ p(t) = p_o + p_1x + p_2x^2.$
$T(λp) = T(λp_o + λp_1x + p_2x^2) = λp(1) - λp(0) = λ(p(1) - p(0)) = λT(p) $ as required.
Would this be correct?
linear-algebra functions
edited Aug 28 at 14:16
amWhy
190k26221433
asked Aug 28 at 13:43
Randy Rogers
726
edited Aug 28 at 14:16
amWhy
190k26221433
edited Aug 28 at 14:16
amWhy
190k26221433
edited Aug 28 at 14:16
amWhy
190k26221433
190k26221433
asked Aug 28 at 13:43
Randy Rogers
726
asked Aug 28 at 13:43
Randy Rogers
726
asked Aug 28 at 13:43
Randy Rogers
726
726
1
It's fine, but you don't need to write down the express form of each polynomial.
– DonAntonio
Aug 28 at 13:46
welcome to MSE. please try avoid using in embed pictures, but instead try using MathJax
– Jneven
Aug 28 at 14:16
Sure, but whats wrong with using embed pictures? It saves me time and i have alot of studying to do @Jneven
– Randy Rogers
Aug 28 at 14:18
other users won't be able to find the question if they are looking for it. this is a community. asking questions is a big part of it. but the information should be as available as possible.
– Jneven
Aug 28 at 14:20
1
Randy, this site is not for you, it's for everyone. Using text and MathJax instead of pictures improves the overall interface of the site, the readability for people who use screen readers, and search results both on the site and from Google when people are looking for related topics and questions. Remember that the questions you ask are likely to be similar to questions that other people have elsewhere. Not everything is confined around your exams.
– Asaf Karagila♦
Aug 28 at 15:19
 |Â
show 4 more comments
1
It's fine, but you don't need to write down the express form of each polynomial.
– DonAntonio
Aug 28 at 13:46
welcome to MSE. please try avoid using in embed pictures, but instead try using MathJax
– Jneven
Aug 28 at 14:16
Sure, but whats wrong with using embed pictures? It saves me time and i have alot of studying to do @Jneven
– Randy Rogers
Aug 28 at 14:18
other users won't be able to find the question if they are looking for it. this is a community. asking questions is a big part of it. but the information should be as available as possible.
– Jneven
Aug 28 at 14:20
1
Randy, this site is not for you, it's for everyone. Using text and MathJax instead of pictures improves the overall interface of the site, the readability for people who use screen readers, and search results both on the site and from Google when people are looking for related topics and questions. Remember that the questions you ask are likely to be similar to questions that other people have elsewhere. Not everything is confined around your exams.
– Asaf Karagila♦
Aug 28 at 15:19
1
1
It's fine, but you don't need to write down the express form of each polynomial.
– DonAntonio
Aug 28 at 13:46
It's fine, but you don't need to write down the express form of each polynomial.
– DonAntonio
Aug 28 at 13:46
welcome to MSE. please try avoid using in embed pictures, but instead try using MathJax
– Jneven
Aug 28 at 14:16
welcome to MSE. please try avoid using in embed pictures, but instead try using MathJax
– Jneven
Aug 28 at 14:16
Sure, but whats wrong with using embed pictures? It saves me time and i have alot of studying to do @Jneven
– Randy Rogers
Aug 28 at 14:18
Sure, but whats wrong with using embed pictures? It saves me time and i have alot of studying to do @Jneven
– Randy Rogers
Aug 28 at 14:18
other users won't be able to find the question if they are looking for it. this is a community. asking questions is a big part of it. but the information should be as available as possible.
– Jneven
Aug 28 at 14:20
other users won't be able to find the question if they are looking for it. this is a community. asking questions is a big part of it. but the information should be as available as possible.
– Jneven
Aug 28 at 14:20
1
1
Randy, this site is not for you, it's for everyone. Using text and MathJax instead of pictures improves the overall interface of the site, the readability for people who use screen readers, and search results both on the site and from Google when people are looking for related topics and questions. Remember that the questions you ask are likely to be similar to questions that other people have elsewhere. Not everything is confined around your exams.
– Asaf Karagila♦
Aug 28 at 15:19
Randy, this site is not for you, it's for everyone. Using text and MathJax instead of pictures improves the overall interface of the site, the readability for people who use screen readers, and search results both on the site and from Google when people are looking for related topics and questions. Remember that the questions you ask are likely to be similar to questions that other people have elsewhere. Not everything is confined around your exams.
– Asaf Karagila♦
Aug 28 at 15:19
 |Â
show 4 more comments
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1
It's fine, but you don't need to write down the express form of each polynomial.
– DonAntonio
Aug 28 at 13:46
welcome to MSE. please try avoid using in embed pictures, but instead try using MathJax
– Jneven
Aug 28 at 14:16
Sure, but whats wrong with using embed pictures? It saves me time and i have alot of studying to do @Jneven
– Randy Rogers
Aug 28 at 14:18
other users won't be able to find the question if they are looking for it. this is a community. asking questions is a big part of it. but the information should be as available as possible.
– Jneven
Aug 28 at 14:20
1
Randy, this site is not for you, it's for everyone. Using text and MathJax instead of pictures improves the overall interface of the site, the readability for people who use screen readers, and search results both on the site and from Google when people are looking for related topics and questions. Remember that the questions you ask are likely to be similar to questions that other people have elsewhere. Not everything is confined around your exams.
– Asaf Karagila♦
Aug 28 at 15:19