110年 - 110 國立清華大學碩士班考試入學試題_數學系碩士班:線性代數#105741

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 L_A denotes the linear transformation from Fⁿ to F^m 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 that

You 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 A^tx = 0 has no nonzero solutions.

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

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

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.