1Give a mathematical definition of what it means for H of Rn

1)Give a mathematical definition of what it means for H of R^n to a subspace of R^n?

2)Give a mathematical definition of linear transformation.

3)Give a mathematical definition of what it meant by S = span{v1, v2, . . . , vp}

4)Is this statement true? If A is invertible then columns of A1 are linearly independent.

Solution

1. A non empty set H of R^n that is closed under both vector addition as well as scalar multipliction is called a subspace of R^n.

Closed under addition

if x, y is in H then x+y is also be in H

Closed under scalar multiplication

if kx is in H for all vectors x is in H and k is in R