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1.). 2.) If T is a linear mapping such that the dimension of the null space of T is greater than zero, then for every vector in the range of T the pre-image of is a coset of the null space of T. Give an example to illustrate this concept. 3.).

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If T is a linear mapping such that the dimension of the null space

of T is greater than zero, then for every vector in the range

of T the pre-image of is a coset of the null space of T.

Give an example to illustrate this concept.

If A is similar to B , is A necessarily similar to AB?

either a proof or a counterexample.

Find eigenvalues and eigenvectors for A , and if possible

a diagonal matrix that is similar to A.

Find eigenvalues and eigenvectors for each of the following:

Give an example of a matrix A such that:

M=

Give at least one eigenvector that belongs to the null space of M.

Give at least one eigenvector that does NOT belong

to the null space of M. Give the corresponding eigenvalue.

A is an nn matrix.

aik represents the element in row i column k.

aik = (-1)i + k for all i and k.

Give an example of an eigenvalue that is different from 0

and describe an eigenvector to go with it.

60°

10.)

The vectors and form a 60 degree angle.

b = ?