The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z + 4x²is -8- 8 in P₂. The standard basis polynomials for this space are {1, z, z²}. represented by the vector The function F, defined by F(p(x)) = (x + 3). takes the derivative of p(x) and then multiplies the result by (z + 3). a) Write the matrix M for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.] M= b) The number 0 is an eigenvalue for this transformation. Draw three different non-zero polynomials in P₂ that are eigenvectors corresponding to λ = 0. Hint -6-5-4-3-2 Clear All Draw: Hint -2 Hint c) The number 1 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are eigenvectors corresponding to λ = 1. Clear All Draw: -6-5-4-3 4 -2 -3. 0 5 4- 3 ist 1 -1 -4 -5. -6+ 6 4 3- 2- 1 -1 -3 -4 -5 -6 d) The number 2 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are eigenvectors corresponding to A = 2. /^ 6 5- 4 d p(x), is a linear transformation from P₂ to P2. It 3 2 1 41 -2 -3. -4- -5- for to ++▬▬▬▬▬▬▬▬▬▬▬▬▬▬ ▬▬▬▬
The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z + 4x²is -8- 8 in P₂. The standard basis polynomials for this space are {1, z, z²}. represented by the vector The function F, defined by F(p(x)) = (x + 3). takes the derivative of p(x) and then multiplies the result by (z + 3). a) Write the matrix M for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.] M= b) The number 0 is an eigenvalue for this transformation. Draw three different non-zero polynomials in P₂ that are eigenvectors corresponding to λ = 0. Hint -6-5-4-3-2 Clear All Draw: Hint -2 Hint c) The number 1 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are eigenvectors corresponding to λ = 1. Clear All Draw: -6-5-4-3 4 -2 -3. 0 5 4- 3 ist 1 -1 -4 -5. -6+ 6 4 3- 2- 1 -1 -3 -4 -5 -6 d) The number 2 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are eigenvectors corresponding to A = 2. /^ 6 5- 4 d p(x), is a linear transformation from P₂ to P2. It 3 2 1 41 -2 -3. -4- -5- for to ++▬▬▬▬▬▬▬▬▬▬▬▬▬▬ ▬▬▬▬
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z + 4x²is
-8-
8 in P₂. The standard basis polynomials for this space are {1, 2, z²}.
represented by the vector
The function F, defined by F(p(z)) = (x+3).
takes the derivative of p(x) and then multiplies the result by (z + 3).
a) Write the matrix M for this linear transformation according to the standard basis polynomials. [Hint:
Find where the standard basis polynomials go under this transformation.]
M=
b) The number 0 is an eigenvalue for this transformation. Draw three different non-zero polynomials in P₂
that are eigenvectors corresponding to λ = 0.
Hint
-6-5-4-3-2
Clear All Draw:
Hint
-2
Hint
4
-2
-3.
c) The number 1 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are
eigenvectors corresponding to λ = 1.
Clear All Draw:
-6 -5 -4 -3
0
5
4-
3
1
-1
-4
-5.
-6+
6
4
3-
2-
1
-1
-3
-4
-5
-6
d
P(2), is a linear transformation from P₂ to P₂. It
/^
d) The number 2 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are
eigenvectors corresponding to A = 2.
6
5-
4
3
2
1
41
-2
-3
-4-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab289986-80a9-4865-81ee-c0ba3a526505%2F9409ea3c-1f96-41e4-8e39-6277a1a79b33%2Fh2qybk_processed.png&w=3840&q=75)
Transcribed Image Text:The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z + 4x²is
-8-
8 in P₂. The standard basis polynomials for this space are {1, 2, z²}.
represented by the vector
The function F, defined by F(p(z)) = (x+3).
takes the derivative of p(x) and then multiplies the result by (z + 3).
a) Write the matrix M for this linear transformation according to the standard basis polynomials. [Hint:
Find where the standard basis polynomials go under this transformation.]
M=
b) The number 0 is an eigenvalue for this transformation. Draw three different non-zero polynomials in P₂
that are eigenvectors corresponding to λ = 0.
Hint
-6-5-4-3-2
Clear All Draw:
Hint
-2
Hint
4
-2
-3.
c) The number 1 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are
eigenvectors corresponding to λ = 1.
Clear All Draw:
-6 -5 -4 -3
0
5
4-
3
1
-1
-4
-5.
-6+
6
4
3-
2-
1
-1
-3
-4
-5
-6
d
P(2), is a linear transformation from P₂ to P₂. It
/^
d) The number 2 is an eigenvalue for this transformation. Draw three different polynomials in P₂ that are
eigenvectors corresponding to A = 2.
6
5-
4
3
2
1
41
-2
-3
-4-
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