- #1
says
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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 ]