Question
Consider the basis e1 = (−2, 4, −1), e2 = (−1, 3, −1) and e3 = (1, −2, 1) of R3
over R. Find the dual basis of {e1, e2, e3}.
Answer :
Word Count : 638
__________ ______ __________ ___ ________ _________ _________ _____.
________ _______ ______ ___ ______ ____ __________ ___ ______ ___ ________ __________.
_______ ___ ___ _______ ____ ____.
__________ ____ __________ _______ ____ _________ ____ __________ ________ _________ _____.
_________ ______ ________ ____ __________ ________ __________ ________ ____.
_________ _______ _______ _______ _______ ___ ____ _________.
________ _______ ____ ________ _________ __________ _______ _______ _________ ______ ________.
__________ __________ ___ __________ ___ ___ __________ ___ __________ _______ _________ _____.
_____ ______ ____ _______ ________ _____ _____ _______ ______ _____ ___.
______ ________ _______ _______ ______ ____ ____ _____.
____ _________ ___ _______ _______ _____.
_________ ______ ____ ____ ________.
__________ ________ ___ ______ _________ _______ ________ __________ _________ ______ ___.
_____ _____ ______ __________ _____ ___ __________ ___ __________ ____ ___ ______.
______ ____ ______ ____ _________ ____ _____ _____.
_____ ________ ______ _____ _______ __________ ________ ______ ______ _____.
_______ ________ _______ _________ _________ _______ ________ ___ __________ ___ _____.
____ ____ ________ ___ _______ _______.
_________ ________ ____ _________ ___ ________ ________.
____ _________ _________ ___ ________ ___ ________ _______ ___ _____ ____.
________ _______ _______ _________ _____ ________ ___ _______ ______ ______.
____ _____ __________ _______ ________ __________ _____ _____.
___ _____ _____ ________ ____.
__________ _____ _______ ________ ________ _________ _______ ________ _______.
___ _____ __________ _______ ___ ___.
____ ___ _______ _____ _______ _________ _____ ____ _____ _________ __________ __________.
____ ___ _____ ___ ___ _______ _______ ___ _______ ________ ___.
_________ ____ ____ _________ _______ ___.
_________ ____ _____ _____ _________ _____.
______ _____ ___ ______ ________ ________ ___.
_________ _____ ___ _____ ______ ____ ___ _________.
__________ ________ __________ ________ _____ ______ __________ _____.
_______ ______ _______ ___ ____ _______ ____.
_________ ______ _______ ________ _____ __________ _____.
______ _____ ______ ___ ______ __________.
_____ ___ ____ ______ _______ _______ __________ ___.
____ ______ ___ ___ ______ _____ ______ ___ ______.
_____ ____ ___ ___ _____.
__________ ________ ______ ________ ________ _______ _________ __________ ____ ____.
________ ___ _______ ___ _______ _______ __________ ______ ________ ___ _____ ___.
_________ ___ _______ ________ ________ ___.
________ ____ ________ _______ ____ ________ ___.
_____ __________ ____ _______ __________ ___ ____ _______.
_____ _________ ________ ______ ____ ______.
________ ________ ___ _______ _________ ________ _______ _______ _______ ___ _____ _________.
______ _________ ________ ___ ______ _______ ____ ____ ____.
____ ____ _______ __________ _________ ____.
__________ ______ _________ _______ _______ ____ ____ _______ __________ _________ ____ _______.
____ ________ ______ ______ ______ _______.
________ ________.
Skip in 10
To find the dual basis of the given set of vectors \( \{e_1, e_2, e_3\} \), we need to follow these steps: ### 1. Understand the problem: We are given the basis vectors: \[ e_1 = (-2, 4, -1), \quad e_2 = (-1, 3, -1), \quad e_3 = (1, -2, 1) \] These are vectors in \( \mathbb{R}^3 \), and we need to find the dual basis \( \{e_1^*, e_2^*, e_3^*\} \), which is the set of linear functionals that satisfy the following conditions: \[ e_i^*(e_j) = \delta_{ij} \] where \( \delta_{ij} \) is the Kronecker delta, which is 1 if \( i = j \) and 0 otherwise. ### 2. Setup the system of equations: The dual basis \( \{e_1^*, e_2^*, e_3^*\} \) must satisfy the condition: \[ e_i^*(e_j) = \delta_{ij} \] This means that the functional \( e_1^* \) should pick out the first component of \( e_1 \), and similarly for \( e_2^* \) and \( e_3^* \). Let: \[ e_1^* = (a_1, b_1, c_1), \quad e_2^* = (a_2, b_2, c_2), \quad e_3^* = (a_3, b_3, c_3) \] Now, we need to solve for the coefficients \( a_1, b_1, c_1, \dots, a_3, b_3, c_3 \) such that the conditions \( e_i^*(e_j) = \delta_{ij} \) are satisfied. ### 3.
___ ______ ____ _________ _______ _____ _____ _________ __________ __________.__________ ______ __________ ___ ________ _________ _________ _____.
________ _______ ______ ___ ______ ____ __________ ___ ______ ___ ________ __________.
_______ ___ ___ _______ ____ ____.
__________ ____ __________ _______ ____ _________ ____ __________ ________ _________ _____.
_________ ______ ________ ____ __________ ________ __________ ________ ____.
_________ _______ _______ _______ _______ ___ ____ _________.
________ _______ ____ ________ _________ __________ _______ _______ _________ ______ ________.
__________ __________ ___ __________ ___ ___ __________ ___ __________ _______ _________ _____.
_____ ______ ____ _______ ________ _____ _____ _______ ______ _____ ___.
______ ________ _______ _______ ______ ____ ____ _____.
____ _________ ___ _______ _______ _____.
_________ ______ ____ ____ ________.
__________ ________ ___ ______ _________ _______ ________ __________ _________ ______ ___.
_____ _____ ______ __________ _____ ___ __________ ___ __________ ____ ___ ______.
______ ____ ______ ____ _________ ____ _____ _____.
_____ ________ ______ _____ _______ __________ ________ ______ ______ _____.
_______ ________ _______ _________ _________ _______ ________ ___ __________ ___ _____.
____ ____ ________ ___ _______ _______.
_________ ________ ____ _________ ___ ________ ________.
____ _________ _________ ___ ________ ___ ________ _______ ___ _____ ____.
________ _______ _______ _________ _____ ________ ___ _______ ______ ______.
____ _____ __________ _______ ________ __________ _____ _____.
___ _____ _____ ________ ____.
__________ _____ _______ ________ ________ _________ _______ ________ _______.
___ _____ __________ _______ ___ ___.
____ ___ _______ _____ _______ _________ _____ ____ _____ _________ __________ __________.
____ ___ _____ ___ ___ _______ _______ ___ _______ ________ ___.
_________ ____ ____ _________ _______ ___.
_________ ____ _____ _____ _________ _____.
______ _____ ___ ______ ________ ________ ___.
_________ _____ ___ _____ ______ ____ ___ _________.
__________ ________ __________ ________ _____ ______ __________ _____.
_______ ______ _______ ___ ____ _______ ____.
_________ ______ _______ ________ _____ __________ _____.
______ _____ ______ ___ ______ __________.
_____ ___ ____ ______ _______ _______ __________ ___.
____ ______ ___ ___ ______ _____ ______ ___ ______.
_____ ____ ___ ___ _____.
__________ ________ ______ ________ ________ _______ _________ __________ ____ ____.
________ ___ _______ ___ _______ _______ __________ ______ ________ ___ _____ ___.
_________ ___ _______ ________ ________ ___.
________ ____ ________ _______ ____ ________ ___.
_____ __________ ____ _______ __________ ___ ____ _______.
_____ _________ ________ ______ ____ ______.
________ ________ ___ _______ _________ ________ _______ _______ _______ ___ _____ _________.
______ _________ ________ ___ ______ _______ ____ ____ ____.
____ ____ _______ __________ _________ ____.
__________ ______ _________ _______ _______ ____ ____ _______ __________ _________ ____ _______.
____ ________ ______ ______ ______ _______.
________ ________.
IGNOU ACADEMY
Your IGNOU journey, made easier.
- Solved Assignments
- Study Notes
- Guess Papers
- Question Papers
Continue Reading — It's Free
Sign in with Google to unlock more of this answer.
✓ Free access
✓ Unlock up to 60% of the answer
✓ No payment required
✓ Unlock up to 60% of the answer
✓ No payment required
Quick Google sign-in. No separate registration form.
Related Questions
Complete the set S = {x3+x2+1,x2+x+1,x+1} to get a basis of P3
Find the radius and the center of the circular section of the sphere |r| = 26 cut off by the plane r . (2i+6j+3k) = 70.
Let T : R3 → R3 be defined by
T (x1, x2, x3) = (x1 − x3, x2 − x3, x1) .
Is T invertible? If yes, find a rule for T −1 like the one which defines T .
Define T : R3 → R3 by
T (x, y, x) = (x + y, y, 2x − 2y + 2z).
Check that T satisfies the polynomial (x − 1)2(x − 2). Find the minimal polynomial of T .
Reduce the conic x2+6xy+y2 - 8 = 0 to standard form. Hence identify the given conic.
Find the inverse of the matrix
using row reduction.
- Consider the linear operator T : C3 → C3, defined by
T (z1, z2, z3) = (z1 − iz2, iz1 + 2z2 + iz3, −iz2 + z3) .
- Compute T ∗ and check whether T is self-adjoint.
ii) Check whether T is unitary