Find the matrix of the linear transformation with respect to the ordered basis{(0, −1), (−1, 0)}.
We are given a linear transformation $$ T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \text{ defined by } T(x, y) = (2y, x - y) $$ and we are to find the matrix of $T$ with respect to the ordered basis $$ B = \{ \mathbf{v}_1 = (0, -1),\ \mathbf{v}_2 = (-1, 0) \}. $$ --- ### Step 1: Apply $T$ to each basis vector #### ____ _____ ____ ______ ________ _________ __________.
_____ ___ ______ ______ _________ ______ ____ __________.
_______ ___ __________ ___ _______ ____ ________.
_______ ___ ______ _____ ___ ___ ________ _________ ______.
__________ ___ ________ ____ ___ _________ _______.
___ ______ _____ __________ ____ _________ ________.
_________ ________ __________ _______ ________ _______ ____ _____ ______ ____ ________.
___ _______ ____ ______ ___ _________.
___ _______ _________ ____ _________.
___ _____ ____ ___ ______ ______ _______ ___ _________ ____ ______ _________.
_______ _______ ____ __________ _____ _________.
____ ______ ______ _____ _________ _________ _________ ________ _________.
____ ____ ______ ________ ______.
________ ____ ________ _________ ___ _______ _______ __________.
__________ ______ ________ _______ _____ ____.
_________ ___ _________ _______ _________ ________ _____ ___ _______ ________ ____ ______.
_____ ____ _____ _____ ____ _______ ___ ______.
_____ _____ _______ ____ ________ _____ ________ _________.
____ ___ ___ ____ ________ _________ _____.
_____ ____ ________ __________ ______ __________ __________ ______.
____ ______ ____ _________ ________.
___ _______ ____ _________ _____ ________.
_________ ________ _____ _____ _________ ___.
_____ ___ ________ ____ _____.
______ ________ _____ ________ _______ __________ ______ _______ __________ _____ _____ ______.
______ __________ _________ _____ _____ ____ __________ _______ _____ _____ _______.
____ _______ ___ __________ ___ _______.
______ _______ _______ _________ ____ ____.
___ _____ _________ ___ __________ _____ ___ ______ _____ __________ ______.
___ _______ __________ ___ _____ _______ ___ __________ ______ _______.
_________ ____ ______ _____ ______ __________ ________ _____ _____ ___ ______.
_______ ___ _______ _________ __________ _______ ____ _______ _____ ___ ______.
_______ _____ _________.
Get Full Answer on WhatsApp
If {?1 , ?2 } is an ordered basis of ℝ 2 and {?1 (?) , ?2 (?)} is the corresponding dual basis find
Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample
i) If ?1 and ?2 are proper subspaces of a non-zero, finite dimensional, vector space ? and
ii) If ? is a vector space and is such that ?? ≠ ?? if ? ≠ ?, then S
is a linearly independent set
iii) ? are linear operators on a finite dimensional vector space ? and ?1 ∘ ?2 is invertible, ?2 ∘ ?1 is also invertibl
iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial
v) If ?1 , ?2 ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?1 + ?2 is also a self adjoint operator.
Find the kernel of the linear transformation ? ∶ ℝ2 → ℝ2 defined by
Check whethe is an eigenvector for the matrix
What is the corresponding eigenvalue?
Find the eigenvalues and eigenvectors of the matrix Is the matrix diagonalisable? Justify your answer.
Check whetherr ? ∶ ℝ2 → ℝ2, defined by ? (?, ?) = (−?, ?) is a linear transformation.
Consider the linear operator ? ∶ ℂ3 → ℂ3 , defined by
i) Compute ? ∗ and check whether ? is self-adjoint
ii) Check whether ? is unitary.