Reference no: EM132344830
Assignment -
Q1. For the following -
Define the linear transformation T by T(x) = Ax. Find
(a) ker(T),
(b) nullity(T),
(c) range(T), and
(d) rank(T).
Q2. For the following find the nullity of.
T: R4 → R2, rank (T) = 2
T: P5 → P2, rank (T) = 3.
Q3. For the following matrices find:
a. the characteristic equation and
b. the eigenvalues (and corresponding eigenvectors) of the matrix.
Q4. For the following consider the linear transformation whose matrix relative to the standard basis is given. Find (a) the eigenvalues of,(b) a basis for each of the corresponding eigenspaces, and (c) the matrix for relative to the basis, where is made up of the basis vectors found in part (b).
Q5. Proof - prove that λ = 0 is an eigenvalue of A if and only if A is singular.
Q6. Proof - For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1?
Q7. Prove that A and AT have the same eigenvalues. Are the eigenspaces the same?
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