- #1

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## Homework Statement

Given the transformation fh : R 3 → R 3 defined by fh(x, y, z) = (x−hz, x+y −hz, −hx+z), where h ∈ R is a parameter.

a) Find, for all possible values of h, Ker(fh), Im(fh), their bases and dimensions.

b) Is fh an isomorphism for some value of h?

## Homework Equations

Ax=o

## The Attempt at a Solution

[ x−hz ]

[ x+y −hz ]

[ −hx+z ]

Matrix associated to the linear transformation =

[ 1 0 -h ]

[ 1 1 -h ]

[ -h 0 1 ]

Reduced row echelon form =

[ 1 0 -h ]

[ 0 1 0 ]

[ -h 0 1 ]

x - hz = 0

y=0

-hx+z=0

Therefore h = 1.

ker(fn)

[ 1 0 -1 ]

[ 0 1 0 ] = 0

[ -1 0 1 ]

ker(fn) = span { (1,0,-1) , (-1,0,1) } = range(ker(fn))

To find the Im(fn) we will firstly set up the original matrix with h=1

[ 1 0 -1 ]

[ 1 1 -1 ]

[ -1 0 1 ]

Then transpose =

[ 1 1 -1 ]

[ 0 1 0 ]

[ -1 -1 1 ]

Row reduced echelon form of the transposed matrix =

[ 1 0 -1 ]

[ 0 1 0 ]

[ 0 0 0 ]

Im(fn) = span { (1,0,-1) }

Dimension of the ker(fn) and Im(fn) = 1

Because this is the only vector in the Im(fn) it also = the range(fn). I'm still a bit confused about what the range of a linear transformation is and how to find it.

I'm not really sure where to start with part b) of the question...

The only time the matrix will = 0 is when x,y,z = 0 (trivial solution) so I would say it is an isomorph, but I'm not totally sure.

[ 1 0 -1 ]

[ 1 1 -1 ]

[ -1 0 1 ]