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inner
Recent questions tagged inner
63
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1
answers
Show $L$ is not positive definite by proving $0$ is an eigenvalue of $L$.
Let $L = D - W \in \mathbb{R}^{n \times n}$ be the graph Laplacian for data with an associated symmetric weight matrix $W$, and $w_{ij} \in [0,1]$ for all $i,j = 1,...,n$...
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113k
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Mar 10
Mathematics
positive
definite
prove
product
inner
properties
matrix
+
–
62
views
1
answers
Prove that $\langle x,y\rangle_M = x M y^T$ satisfies the properties of an inner product if $M$ is positive definite.
(a) Prove that $\langle x,y\rangle_M = x M y^T$ satisfies the properties of an inner product if $M$ is positive definite.(b) Show that $\langle x,y\rangle_M$ need not be ...
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Platinum
113k
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asked
Mar 10
Mathematics
prove
properties
inner
product
positive
definite
+
–
60
views
1
answers
Prove that the Euclidean dot product . . .
Prove that the Euclidean dot product $\langle x, y \rangle = \sum_{i=1}^n x_i y_i, x, y \in \mathbb{R}^n$ is an inner product, where an inner product is a binary function...
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Platinum
113k
points
MathsGee
asked
Mar 10
Mathematics
product
vectors
euclidean
inner
dot
two
prove
+
–
400
views
1
answers
A vector in standard form has components \(\langle 3,10\rangle\). What is the initial point?
A vector in standard form has components \(\langle 3,10\rangle\). What is the initial point?
Edzai Zvobwo
Bronze Status
6.2k
points
Edzai Zvobwo
asked
Jul 16, 2023
Mathematics
vector
product
space
closed
orbit
fields
inner
+
–
338
views
1
answers
Was the Pythagorean School a cult?
Was the Pythagorean School a cult?
MathsGee
Platinum
113k
points
MathsGee
asked
Feb 8, 2023
Mathematics
pythagorean
theorem
hypotenuse
triangle
generalize
arbitrary
inner
+
–
326
views
1
answers
What was the significance of the invention of logarithms?
What was the significance of the invention of logarithms?
AstraNova
Diamond
68.1k
points
AstraNova
asked
Feb 6, 2023
Mathematics
theorem
hypotenuse
triangle
generalize
arbitrary
inner
pythagorean
+
–
510
views
1
answers
What was the significance of the Pythagorean Theorem?
What was the significance of the Pythagorean Theorem?
AstraNova
Diamond
68.1k
points
AstraNova
asked
Feb 6, 2023
Mathematics
theorem
pythagorean
hypotenuse
triangle
generalize
arbitrary
inner
+
–
313
views
1
answers
What is the dot product of two vectors?
What is the dot product of two vectors?
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jan 27, 2023
Mathematics
product
dot
vectors
inner
euclidean
two
knowing
+
–
609
views
1
answers
What is the Pythagorean Theorem?
What is the Pythagorean Theorem?
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jan 27, 2023
Mathematics
theorem
pythagorean
hypotenuse
triangle
generalize
arbitrary
inner
+
–
381
views
0
answers
What is the Pythagorean theorem?
What is the Pythagorean theorem?
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jan 27, 2023
Mathematics
theorem
pythagorean
hypotenuse
triangle
generalize
arbitrary
inner
+
–
239
views
0
answers
Multiply the polynomials $(x+2)(x+3)$ and $(x-1)(x-4)$.
Multiply the polynomials $(x+2)(x+3)$ and $(x-1)(x-4)$.
MathsGee
Platinum
113k
points
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asked
Jan 9, 2023
Mathematics
polynomials
root
degree
monic
function
inner
product
+
–
217
views
0
answers
Calculate the product of \(2 x-1\) and \(x^2+2 x-3\).
Calculate the product of \(2 x-1\) and \(x^2+2 x-3\).
MathsGee
Platinum
113k
points
MathsGee
asked
Nov 27, 2022
Mathematics
product
calculate
inner
function
graph
expression
equation
+
–
396
views
0
answers
Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that
Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that (i) if \(x<y\), then \(f(x)<f(y)\)(ii) \(f(y f(x))=x^2 f(x y)\) for all \(x, y \in \mathbb{N}\).
MathsGee
Platinum
113k
points
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asked
Sep 23, 2022
Mathematics
set
vector
function
product
space
inner
metric
+
–
512
views
0
answers
Find all functions \(f\) such that \[ f(f(x)+y)=f\left(x^2-y\right)+4 y f(x) \] for all real numbers \(x\) and \(y\).
Find all functions \(f\) such that\[f(f(x)+y)=f\left(x^2-y\right)+4 y f(x)\]for all real numbers \(x\) and \(y\).
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 23, 2022
Mathematics
function
set
vector
product
space
inner
metric
+
–
414
views
0
answers
Find all functions \(f\) from \(\mathbb{R} \backslash\{0\}\) to itself satisfying \[ f(x)+f(y)=f(x y f(x+y)) \] for all \(x, y \neq 0\) with \(x+y \neq 0\).
Find all functions \(f\) from \(\mathbb{R} \backslash\{0\}\) to itself satisfying\[f(x)+f(y)=f(x y f(x+y))\]for all \(x, y \neq 0\) with \(x+y \neq 0\).
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 23, 2022
Mathematics
function
set
vector
space
product
inner
metric
+
–
591
views
0
answers
Find all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) such that
Find all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) such that\[f(x+y)+f(x-y)=2 f(x)+2 f(y) \quad \text { for all } x, y \in \mathbb{Q}\]
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 22, 2022
Mathematics
vector
set
space
product
function
inner
metric
+
–
472
views
0
answers
Determine all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) such that \[ f(x f(y))=(1-y) f(x y)+x^2 y^2 f(y) \] holds for all real numbers \(x\) and \(y\).
Determine all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) such that\[f(x f(y))=(1-y) f(x y)+x^2 y^2 f(y)\]holds for all real numbers \(x\) and \(y\).
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 22, 2022
Mathematics
set
vector
function
space
product
inner
metric
+
–
397
views
0
answers
Find all functions \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) whose minimum value is 1 and \(f(f(n))=f(n)+1 \quad\) for all \(n \in \mathbb{N}_0\)
Find all functions \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) whose minimum value is 1 and \(f(f(n))=f(n)+1 \quad\) for all \(n \in \mathbb{N}_0\)
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 22, 2022
Mathematics
function
set
vector
space
product
inner
metric
+
–
539
views
0
answers
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for all \(x, y\) \[ f\left(x^2\right)-f\left(y^2\right)=(x+y)(f(x)-f(y)) . \]
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for all \(x, y\)\[f\left(x^2\right)-f\left(y^2\right)=(x+y)(f(x)-f(y)) .\]
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 22, 2022
Mathematics
function
set
vector
product
space
inner
metric
+
–
340
views
0
answers
Does there exist a function \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) such that
Does there exist a function \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) such that\[\underbrace{f(f(\ldots f}_{2003}(n) \ldots))=5 n \quad \text { for all } n \in \mathbb...
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 22, 2022
Mathematics
set
vector
function
product
space
inner
metric
+
–
194
views
0
answers
Find all functions \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) satisfying \(f(2)=2, f(n)<f(n+1)\) and \(f(m n)=\) \(f(m) f(n)\) for all \(m, n \in \mathbb{N}\)
Find all functions \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) satisfying \(f(2)=2, f(n)<f(n+1)\) and \(f(m n)=\) \(f(m) f(n)\) for all \(m, n \in \mathbb{N}\)
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 22, 2022
Mathematics
space
function
vector
set
product
inner
metric
+
–
333
views
1
answers
How do I round real numbers to an appropriate degree of accuracy?
How do I round real numbers to an appropriate degree of accuracy?
MathsGee
Platinum
113k
points
MathsGee
asked
Sep 7, 2022
Mathematics
real
numbers
fact
rational
irrational
vectors
inner
+
–
585
views
1
answers
Let \(\mathrm{V}\) be the real vector space of all real \(2 \times 3\) matrices, and let \(\mathrm{W}\) be the real vector space of all real \(4 \times 1\) column vectors.
Let \(\mathrm{V}\) be the real vector space of all real \(2 \times 3\) matrices, and let \(\mathrm{W}\) be the real vector space of all real \(4 \times 1\) column vectors...
MathsGee
Platinum
113k
points
MathsGee
asked
Jul 7, 2022
Mathematics
product
inner
vectors
orthogonal
euclidean
theorem
prove
+
–
292
views
1
answers
Prove that \[ \sum_{i} \sum_{j} a_{i j}=-2, \quad \sum_{j} \sum_{i}=0 . \]
Let \(a_{i j}\) be the number in the \(i\) th row and \(j\) th column of the array\[\begin{array}{lllll}-1 & 0 & 0 & 0 & \ldots\end{array}\]\(\frac{1}{2} \quad-1 \quad 0 ...
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jul 5, 2022
Mathematics
vector
set
product
space
function
metric
inner
+
–
222
views
1
answers
Is \(g\) continuous?
Suppose(a) \(\mid f(x, y) \leq 1\) if \(0 \leq x \leq 1,0 \leq y \leq 1\),(b) for fixed \(x, f(x, y)\) is a continuous function of \(y\),(c) for fixed \(y, f(x, y)\) is a...
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jul 5, 2022
Mathematics
function
space
set
product
vector
metric
inner
+
–
525
views
1
answers
On the calculator side of the flight computer, time is always found on which scale(s)?
On the calculator side of the flight computer, time is always found on which scale(s)?A. Outer.B. Inner scale only.C. Inner scale and far inner scale.
pilot_H
Wooden Status
596
points
pilot_H
asked
Jun 19, 2022
General Knowledge
calculator
side
flight
computer
inner
ppl
+
–
264
views
0
answers
Verify that this formula holds for the probabilities in the figure below
It can be shown that for any three events \(A, B\), and \(C\), the probability that at least one of them will occur is given by\[\begin{aligned}P(A \cup B \cup C)=& P(A)+...
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jun 17, 2022
AI & Data Science
verify
product
inner
inequality
cauchy-schwarz
euclidean
holds
+
–
732
views
1
answers
Find \(\operatorname{gcd}(500,222)\) using euclidean algorithm
Find \(\operatorname{gcd}(500,222)\) using euclidean algorithm
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jun 4, 2022
Mathematics
euclidean
algorithm
k-nn
distances
manhattan
product
inner
+
–
686
views
1
answers
What is the product of 2 and 3?
What is the product of 2 and 3?
AstraNova
Diamond
68.1k
points
AstraNova
asked
Jun 3, 2022
Mathematics
product
inner
dot
keywords
identify
report
ads
+
–
391
views
1
answers
What is the dot (Euclidean inner) product of two vectors?
What is the dot (Euclidean inner) product of two vectors?
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 26, 2022
Mathematics
dot
euclidean
inner
product
two
vectors
+
–
1.6k
views
1
answers
How do I calculate the dot product of two vectors?
How do I calculate the dot product of two vectors?
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 26, 2022
Mathematics
calculate
inner
dot
product
vectors
+
–
557
views
1
answers
Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$ \mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0) $$
Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$ \mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0) $$
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 26, 2022
Mathematics
vector
components
dot
inner
euclidean
product
calculate
+
–
354
views
1
answers
What are the algebraic properties of the dot/inner product?
What are the algebraic properties of the dot/inner product?
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 26, 2022
Mathematics
algebraic
properties
dot
inner
product
+
–
850
views
1
answers
Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$ \mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2} $$
Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$ \mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v...
MathsGee
Platinum
113k
points
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asked
Jan 26, 2022
Mathematics
prove
vectors
euclidean
inner
product
+
–
576
views
0
answers
Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).
Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right...
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Platinum
113k
points
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asked
Jan 21, 2022
Mathematics
space
product
inner
vector
function
set
metric
+
–
499
views
0
answers
Find the function \(f \in \operatorname{span}\{1 \sin x, \cos x\}\) that minimizes \(\|\sin 2 x-f(x)\|\), where the norm comes from the inner product
Find the function \(f \in \operatorname{span}\{1 \sin x, \cos x\}\) that minimizes \(\|\sin 2 x-f(x)\|\), where the norm comes from the inner product\[\langle f, g\rangle...
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Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
norm
span
minimize
product
inner
+
–
780
views
0
answers
Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\).
Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=...
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
product
inner
space
vector
function
set
metric
+
–
375
views
0
answers
Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\).
Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\)....
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
inner
product
space
vector
function
metric
set
+
–
586
views
0
answers
Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.
Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.a) Show that \(\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)\) is an inner product for this space.b) U...
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Platinum
113k
points
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asked
Jan 21, 2022
Mathematics
space
inner
product
orthogonal
map
basis
+
–
413
views
0
answers
Using the inner product of the previous problem, let \(\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}\) be an orthogonal basis for the space \(\mathcal{P}_{2}\) of quadratic polynomials and . . .
Using the inner product of the previous problem, let \(\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}\) be an orthogonal basis for the space \(\mathcal{P}_{2}\) of quadratic ...
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Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
inner
product
orthogonal
basis
projection
map
+
–
306
views
1
answers
Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\), for which values of the real constants \(\alpha, \beta, \gamma\) are the quadratic polynomials ...
Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\), for which values of the real constants \(\alpha, \beta, \gamma\) are the quadratic polynomial...
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Platinum
113k
points
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asked
Jan 21, 2022
Mathematics
inner
product
quadratic
polynomials
orthogonal
+
–
486
views
0
answers
In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].
In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previou...
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Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
complex
vector
space
hermitian
inner
product
matrix
+
–
470
views
1
answers
Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).
Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).a) Let \(v, w, z\) be vectors in a real inner product space. If \(\langle v, w\r...
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
product
inner
vectors
orthogonal
euclidean
theorem
prove
+
–
527
views
0
answers
Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\) whose degrees are at most \(n\) equipped with the inner product
Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\)...
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Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
inner
product
space
function
vector
metric
set
+
–
590
views
0
answers
Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x\).
Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\ra...
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
vector
space
product
inner
function
set
metric
+
–
497
views
0
answers
Find all vectors in the plane (through the origin) spanned by \(\mathbf{V}=(1,1-2)\) and \(\mathbf{W}=(-1,1,1)\) that are perpendicular to the vector \(\mathbf{Z}=(2,1,2)\).
Find all vectors in the plane (through the origin) spanned by \(\mathbf{V}=(1,1-2)\) and \(\mathbf{W}=(-1,1,1)\) that are perpendicular to the vector \(\mathbf{Z}=(2,1,2)...
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
vectors
orthogonal
euclidean
product
inner
theorem
+
–
462
views
0
answers
Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).
Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).a) Show that the Pythagorean relation \(\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}\) holds if and only if \(V\) and \(W\) are orthogona...
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Platinum
113k
points
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asked
Jan 21, 2022
Mathematics
pythagorean
relation
orthogonal
product
space
inner
+
–
596
views
1
answers
Let \(W\) be a linear space with an inner product and \(A: W \rightarrow W\) be a linear map whose image is one dimensional (so in the case of matrices, it has rank one).
Let \(W\) be a linear space with an inner product and \(A: W \rightarrow W\) be a linear map whose image is one dimensional (so in the case of matrices, it has rank one)....
MathsGee
Platinum
113k
points
MathsGee
asked
Jan 21, 2022
Mathematics
basis
image
map
inner
product
invertible
matrix
+
–
620
views
0
answers
Let \(\mathcal{P}_{3}\) be the space of polynomials of degree at most 3 anD let \(D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) be the derivative operator.
Let \(\mathcal{P}_{3}\) be the space of polynomials of degree at most 3 anD let \(D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) be the derivative operator.a) Using the...
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Platinum
113k
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asked
Jan 21, 2022
Mathematics
vectors
orthogonal
theorem
inner
prove
linear
+
–
550
views
0
answers
Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .
Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 . a) Find a basis for this space.b) Let \(D: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}\) be the ...
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Jan 21, 2022
Mathematics
linear
vectors
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product
orthogonal
theorem
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