4. (12 points) Let . Show that if rank(AB) = m, then rank(BA) =m.

Notation: R denotes the feld of real numbers; C denotes the field of complex numbers. F denotes an arbitrary field; denotes the set of all m x n matrices with entries in F. If T is a linear transformation, R(T) denotes the range of T, and N(T) denotes the null space of T. denotes the transpose of A, and LA denotes the linear transformation from Fn to Fm that sends each vector  1. (12 points) Let V and W be F-vector spaces, and let T: V →W be a linear transformation. Prove that dim R(T)+dim N(T) = dim V if V is fnite-dimensional.

2. (10 points) Find a matrix such thatYou need to show that the matrix you find has the required properties.

3. (12 points) Let Show that the system of linear equations Ax = b has a solution for all b if and only if the system of linear equations Atx = 0 has no nonzero solutions.

(a) (6 points) State the definition of eigenvectors of T.

(b) (6 points) Give an explicit example of T that has no eigenvectors.

(c) (8 points) Prove that T has an eigenvector if F = C.

6. (10 points) Let is an invertible matrix such that AQ is diagonal, then each column vector of Q is an eigenvector of LA.

7. (12 points) Let T:Rn→ Rn be a linear operator. Show that if T preserves the Euclidean distance between any two poiuts, that is, ||T(u) - T(ข)|| = ||u - u||for any u, , then the matrix representation of T relative to the standard basis is an orthogonal matrix.

8. (12 points) Let be a real symmetric matrix. Show that there exists a real symmetric matrix B such that B3 = A.

Problem 6 :Let A and B be elements in (C). Suppose that AB - BA = c⋅(A - B)  for some non-zero c ∈ C. Prove that there exists an invertible matrix P ∈ (C) such that AP and BP are upper-triangular matrices with the same diagonal entries.

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