We're told, let the transformation of vector x
equal the given formulas,
be a linear transformation from R three to R two,
and we're asked to determine if T is onto and/or one-to-one.
To do this, I'll write T of vector x
in the form of T of vector x equals matrix A times vector x.
It wants you to determine matrix A,
if we write matrix A in row echelon form
or reduced row echelon form, and then determine the pivots.
If there are pivots in every column,
the transformation is one-to-one.
If there are pivots in every row,
the transformation is onto.
So we can write this as T of vector x equals.
Because we have two equations with three unknowns,
we would have a two by three coefficient matrix
times a three by one column matrix where.
Because the variables are x one, x two, and x three,
the column matrix has entries x one, x two, x three,
and the two by three matrix will be the coefficient matrix
where the first row from the first equation
is negative five, three, four,
and the second row from the second equation is 20,
negative 12, negative 16.
Notice now that we have the equation
in the form of T of vector x equals matrix A times vector x,
we know matrix A is the two by three matrix,
where again, the first row is negative five, three, four,
and the second row is 20, negative 12, negative 16.
The next step is to write this matrix
in either row echelon form or reduced row echelon form.
I've already found the reduced row echelon form.
In reduced row echelon form,
the first row is one, negative 3/5, negative 4/5,
and the second row is a row of zeros.
And now we identify the pivots.
Notice for the matrix in reduced row echelon form,
there's only one pivot in row one column one.
Because we do not have a pivot in every column,
the transformation is not one-to-one.
And because we also don't have a pivot in every row,
the transformation is also not onto.
So to answer the question, is T onto, the answer is no,
And is T one-to-one, the answer is also no.
Let's take a look at one more example.
This is the same problem with different formulas.
So again, we're going to write the equation
as T of vector x equals matrix A times vector x.
So we have T of vector x equals the coefficient matrix
times vector x as a three by one column matrix
where the components are x one, x two, x three.
Using the coefficients of the first equation,
the first row is negative one, two, three.
Using the second equation,
the second row is two, negative two, negative seven.
So again, now we know that matrix A,
again, is a two by three matrix
where the first row is negative one, two, three,
and the second row is two, negative two, negative seven.
I've already found the reduced row echelon form.
In reduced row echelon form,
the first row is one, zero, negative four,
and the second row is zero, one, negative 1/2.
We have pivots in row one column one and row two column two.
We do not have pivots in every column,
and therefore the transformation is not one-to-one.
However, we do have pivots in every row,
and therefore the transformation is onto.
So is the transformation onto?
The answer is yes.
Is the transformation one-on-one?
The answer is no.
Because you know the second transformation is onto,
we know the second transformation
has the properties as shown here for onto transformations.
I hope you found this helpful.