Math Is Fun Forum

deepz

Member

Registered: 2007-03-20

Posts: 10

Linear algebra

Hi I need some help with proving if some statements are true or false with brief reasons needed.

1) If A is a square matrix whose entries lie in a field K then the determinant of A cubed is an element of K

2) For every field K and every c that is an element of K, there exists a 4x4 matrix A with entries in K such that det(A)= c

3)R^2 is a subspace of C^2( R is the set of real numbers and C the set of complex numbers)

4) The set of integer vectors Z^3 = {[ x y z] transposed: x,y,z are elements of Z} is a subspace of R^3

Thanks in advance:D

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear algebra

1. This is a rather odd question.  But to answer it, one of the first things you must realize about a field is that it is closed under addition and multiplication.  Now how do you determine what the determinant is?

2.  This is a construction problem.  You are asked to construct a matrix with determinant c.  You need to realize that there are in general going to be many matrices which have determinant c, but you only need one.  So it's probably a good idea to go for the easiest one.  That means you'll typically be using a lot of 0's and 1's.  Remember: every field contains an additive identity (0), and a multiplicative identity (1).

3. A complex number z = a + bi, where a and b are real.

4.  To check a subspace, you must check it is closed under addition and scalar multiplication.  Is it?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."