Linear Algebra Determine whether each set equipped with the

Linear Algebra

Determine whether each set equipped with the given operations is avector space. For those that are not vector spaces identify the vector space axioms that fail.

a). The set of all pairs of real numbers of the form (x,y), where x 0, with the standard operations on R^2.

b). The set of all triples of real numbers with the standard vector addition but with scalar multiplication dened by k(x,y,z) = (k^2x,k^2y,k^2z)

c). The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication

Please show work!

Solution

a)

Not a vector space

(1,3) is in this set

So a scalar multiple of this must also be in the set for it to be vector space

MUltiplying by -1 gives

-(1,3) =(-1,-3) is not in this set

Hence not a vector space

b)

1. Given any such two triples , sum of them is also one such triple hence int he set

2. Given such a triple: (x,y,z) and a scalar, k

k(x,y,z)=(k^2x,k^2y,k^2z) is also in the set

Hence a vector spae

c)

Not a vector space

Let, A be a invertibel 2x2 matrix

Then, -A is also invertible 2x2 matrix.

But, A+(-A)=0 matrix which is not invertible