They are all FALSE please help me find a counterexamaple Sup

They are all FALSE please help me find a counterexamaple

Suppose is an inner product for a vector spiece V. Let ueV. Then S = dollar V: vEV, V horizontal u dollar is a subspace of V. Suppose A is a 3 times 3 matrix with exactly one unione eignvalue. The A is defective. Suppose A is a 8 times 4 matrix with rank (A) = 4. Then colsace (A) = R^4

Solution

5) S is a subspace

if v1,v2 is in S

then <v1, u> = <v2,u> = 0

this implies any linear combination of v1 and v2 is in S

since <a1v1 + a2v2,u> = a1<v,u> + a2<v2,u> = 0

6)this is not necessarily true. Identity matrix is not defective though it has only one eigen value i.e. 1

7) Column space is subspace of R^8 . if rank is 4 then column space has dimension 4. it is not equal to R^4 . though it is isomorphic to R^4.