Reference no: EM132344830
Assignment -
Q1. For the following -
![222_figure.png](https://secure.expertsmind.com/CMSImages/222_figure.png)
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.
![1209_figure1.png](https://secure.expertsmind.com/CMSImages/1209_figure1.png)
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).
![625_figure2.png](https://secure.expertsmind.com/CMSImages/625_figure2.png)
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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