(first three digits), v = 1 4 (last four digits) 6. Let u = %3D %3D 5 2 15 7 4 9 Let A = (place your whole ID in each row) L2 1 5 7 4 9 Calculations. 1. Construct projav. Round each component to four decimal places. 2. Let p = 2+ 3x – x³. Determine the coordinate matrix of p relative to the basis B = {1,1 – x, 1+ x², 1 – x³}. %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let å = 2 (first two ID digits),
(middle three ID digits), 1 (last two digits)
Let u =
(first three digits), v =
(last four digits)
5 7 4
5 7
2 1
6.
Let A =
(place your whole ID in each row)
9.
Calculations.
1. Construct projzv. Round each component to four decimal places.
I
2. Let p = 2+ 3x – x³. Determine the coordinate matrix of p relative
to the basis B = {1,1 – x, 1 + x², 1 – x³}.
3. Construct the orthogonal complement S for S = span{u, v}.
4. Use the Gram-Schmidt Process to construct an orthonormal basis
for the vector space V = span{a, b, c}.
%3D
Round answers to four decimal places, where necessary.
5. Consider the basis B = {(2,-v2). (2, v2)} for R². Define (ỷ, 2) =y1z1 +y2z2.
%3D
a) Verify that B is an orthonormal basis.
b) Calculate [w, for w (20,-6v2).
6. Define a linear transformation T: R → R² by T(x) = Ax.
Calculate the image of (1, 0, 2, 1, 2, -1, 1, 2).
b) Calculate the pre-image of (60, 120).
c) Determine a basis of the range of the transformation.
d) Determine a basis for the kernel of the transformation.
7. Define T: R3 → R³ such that T(a) = (1,2,1), T(b) = (0,1,3),T(C) = (1,0, –1).
Calculate T(-2,3,-1).
8. Define T: R2 → R² by T(x, y) = (3x – 2y, y - 2x).
a) Construct the standard matrix for T.
b) Construct the matrix for T relative to the basis {(5,2), (2,1)}.
c) Construct the matrix for T-1 relative to the basis {(1,1), (1,2)}.
Transcribed Image Text:Let å = 2 (first two ID digits), (middle three ID digits), 1 (last two digits) Let u = (first three digits), v = (last four digits) 5 7 4 5 7 2 1 6. Let A = (place your whole ID in each row) 9. Calculations. 1. Construct projzv. Round each component to four decimal places. I 2. Let p = 2+ 3x – x³. Determine the coordinate matrix of p relative to the basis B = {1,1 – x, 1 + x², 1 – x³}. 3. Construct the orthogonal complement S for S = span{u, v}. 4. Use the Gram-Schmidt Process to construct an orthonormal basis for the vector space V = span{a, b, c}. %3D Round answers to four decimal places, where necessary. 5. Consider the basis B = {(2,-v2). (2, v2)} for R². Define (ỷ, 2) =y1z1 +y2z2. %3D a) Verify that B is an orthonormal basis. b) Calculate [w, for w (20,-6v2). 6. Define a linear transformation T: R → R² by T(x) = Ax. Calculate the image of (1, 0, 2, 1, 2, -1, 1, 2). b) Calculate the pre-image of (60, 120). c) Determine a basis of the range of the transformation. d) Determine a basis for the kernel of the transformation. 7. Define T: R3 → R³ such that T(a) = (1,2,1), T(b) = (0,1,3),T(C) = (1,0, –1). Calculate T(-2,3,-1). 8. Define T: R2 → R² by T(x, y) = (3x – 2y, y - 2x). a) Construct the standard matrix for T. b) Construct the matrix for T relative to the basis {(5,2), (2,1)}. c) Construct the matrix for T-1 relative to the basis {(1,1), (1,2)}.
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