linear algebra help needed!!!

[jessica]

hi finding these questions impossible and was wondering if anyone knew how to answer them!!!....

 

1) a linear transformation T:R^6 -> R^6 is known to have characteristic polynomial

 

x^2(x+1)(x+5)^3

 

determine all possibilities for the minimum polynomial of T

 

2)let V be and inner product space over C with inner product <,> and let u,v be vectors from V which are orthogonal to each other. prove that

 

||au+bv||^2 = |a|^2||u||^2 + |b|^2||v||^2

 

for any complex numbers a and b

 

thank you!!!

[DQW]

1. What do you know about the roots of the minimum polynomial (and the roots of the characteristic polynomial) ?

[DQW]

2. Use the properties of inner products (along with the definition of orthogonality) and expand the LHS. The RHS will follow in just a few steps.

 

additivity : <u,v+w> <u,v> + <u,w>

 

scaling : <au,v> = a <u,v>

 

conjugation : <u,v> = <v,u>*

[jessica]

regarding question (1) dont know anything besides whats in the question, was a past paper question from an exam! i understand how to manipulate question (2) im just not definate about how to deal with the constants a and b when dealin with a complex space!

[matt grime]

if you don't know what the minimal and characteristic poly are (in relation to one another) you shold go and look them up in you course notes.

 

and as i say somewhere else in reply to this, the second part is true becuase of the definitions of an inner product on a complex space, something you need to look up, again in your notes. maths is infinitely easier if you know the definitions.