所屬科目：研究所、轉學考(插大)◆線性代數
1. Does the following statement hold? If it holds, please give aproof. Otherwise, give a counter example.“In every vector space Ｖ over a field Ｆ , and ax=bx implies that a=b”
2. Let , where a and b are both nonzero. Prove that if thetwo eigenvalues are equal, then A is not diagonalizable.
3. Let u, v and w be distinct vectors of a vector space V. Show thatif {u, v, w}is a basis for V , then {u+v+w, v+w, w} is also a basisfor V.
4. Let , find the Jordan canonical form of A.
5. Let where the column vector , Prove that A is symmetric and findthe null space of and its rank.
6. Let and define T=LQ with . Prove that T is a lineartransformation, and write your geometric observation for T.
7. Let and T︰V→V where and .Find the matrix representation [T]B.
8. Let and AB=0. Prove that .
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and ax=bx implies that a=b”
, where a and b are both nonzero. Prove that if thetwo eigenvalues are equal, then A is not diagonalizable.
, find the Jordan canonical form of A.
where the column vector 
, Prove that A is symmetric and findthe null space of and its rank.
and define T=LQ with 
. Prove that T is a lineartransformation, and write your geometric observation for T.
and T︰V→V where 
and 
.Find the matrix representation [T]B.
and AB=0. Prove that 
.