Isomorphism proves: f and g are isomorphisms, prove f + g is an isomorphism

**zeion** · February 13th 2010, 05:30 PM

Question:

Suppose f and g are isomorphisms from U to V. Prove or disprove each of the following statements:
a) The mapping f + g is an isomorphism from U to V

I have no idea where to start. Do I need to show that f(u) + g(u) = (f+g)(u)? Do I need to show that (f+g) is 1-1 and onto?

**Jhevon** · February 13th 2010, 05:47 PM

Originally Posted by zeion

Question:

Suppose f and g are isomorphisms from U to V. Prove or disprove each of the following statements:
a) The mapping f + g is an isomorphism from U to V

I have no idea where to start. Do I need to show that f(u) + g(u) = (f+g)(u)?

no, that's true by definition.

Do I need to show that (f+g) is 1-1 and onto?

yes, and one more thing. $(f + g)(xy) = (f + g)(x) \cdot (f + g)(y)$

i would start with this one first. what can you find?

**zeion** · February 13th 2010, 06:13 PM

Originally Posted by Jhevon

no, that's true by definition.

yes, and one more thing. $(f + g)(xy) = (f + g)(x) \cdot (f + g)(y)$

i would start with this one first. what can you find?

Ok let's try...

Let A be a basis for U and B be a basis for V, and let v belong to V and u belong to V , then
$v = {a_1v_1+...+a_nv_n} , u = {a_1u_1+...+a_nu_n}$
$(f + g)(vu) = (f + g)[(a_1v_1+...+a_nv_n)(a_1u_1+...+a_nu_n)]$
$= (f + g)(a_1+...+a_n)(v_1u_1+...+v_nu_n)$
Does that look right?

**Jhevon** · February 14th 2010, 12:16 AM

Originally Posted by zeion

Ok let's try...

Let A be a basis for U and B be a basis for V, and let v belong to V and u belong to V , then
$v = {a_1v_1+...+a_nv_n} , u = {a_1u_1+...+a_nu_n}$
$(f + g)(vu) = (f + g)[(a_1v_1+...+a_nv_n)(a_1u_1+...+a_nu_n)]$
$= (f + g)(a_1+...+a_n)(v_1u_1+...+v_nu_n)$
Does that look right?

that's not what you were supposed to show. and why are you choosing bases? I suppose this is a linear algebra problem as opposed to an abstract algebra problem(?) if so, then we would use the definition that an isomorphism between two vector spaces is a bijective linear transformation from one to the other. so, is it true that if we take two such transformations, their sum will be such a transformation?

**HallsofIvy** · February 14th 2010, 03:05 AM

Originally Posted by zeion

Question:

Suppose f and g are isomorphisms from U to V. Prove or disprove each of the following statements:
a) The mapping f + g is an isomorphism from U to V

I have no idea where to start. Do I need to show that f(u) + g(u) = (f+g)(u)? Do I need to show that (f+g) is 1-1 and onto?

What kind of structures are U and V? Groups? Rings? Fields? Vector spaces?

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