4 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

... 2. Give an example to show that the union of two subspaces of a vector space V need not be a subspace of V. 3. Prove that any n + 1 vectors in Fn are linearly independent. 4. Define Kernel and Image of a homomorphism T. 5. Define an algebra over a field F. 6. What is eigen value and eigen vector? 7. ...

... 2. Give an example to show that the union of two subspaces of a vector space V need not be a subspace of V. 3. Prove that any n + 1 vectors in Fn are linearly independent. 4. Define Kernel and Image of a homomorphism T. 5. Define an algebra over a field F. 6. What is eigen value and eigen vector? 7. ...

INTRODUCTION TO C* ALGEBRAS - I Introduction : In this talk, we

... is linear and respects the multiplication operator (ie. ψx is a multiplicative linear functional). Furthermore, every multiplicative linear functional is of the form ψx for some x ∈ X. Hence, the set Ω := {ψ : C(X) → C : ψ is multiplicative and linear} can be identified with X. Furthermore, this set ...

... is linear and respects the multiplication operator (ie. ψx is a multiplicative linear functional). Furthermore, every multiplicative linear functional is of the form ψx for some x ∈ X. Hence, the set Ω := {ψ : C(X) → C : ψ is multiplicative and linear} can be identified with X. Furthermore, this set ...

Vector Spaces and Linear Maps

... Proposition 14.17. The standard basis e1 , . . . , en is a basis for F n . (In particular, the empty sequence is a basis for F 0 = {0}.) Exercise 14.18. Find a basis for R2 that contains none of the standard basis vectors, nor any scalar multiple of them. Can you do the same for R3 ? Proposition 14. ...

... Proposition 14.17. The standard basis e1 , . . . , en is a basis for F n . (In particular, the empty sequence is a basis for F 0 = {0}.) Exercise 14.18. Find a basis for R2 that contains none of the standard basis vectors, nor any scalar multiple of them. Can you do the same for R3 ? Proposition 14. ...