Find a matrix transformation mapping \{(1,1,1),(0,1,0),(1,0,2)\} to \{(1,1,1),(0,1,0),(1,0,1)\}

Connor Randall 2022-01-30 Answered
Find a matrix transformation mapping
{(1,1,1),(0,1,0),(1,0,2)}
to
{(1,1,1),(0,1,0),(1,0,1)}
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Expert Answer

Jacob Trujillo
Answered 2022-01-31 Author has 13 answers
Step 1
We wish to find a 3×3 matrix T such that TA=B where
A=[101110102]B=[101110101]
Perhaps the quickest way to find T is to multiply the equation TA=B on the right by A1 to obtain
T=BA1
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mihady54
Answered 2022-02-01 Author has 13 answers
Step 1
The columns of the matrix tell you where it sends the standard basis vectors. For instance if I am interested in the third column then I need to determine what the action of our linear operator is on the column vector ,
[001].
This vector can be written as a linear combination of the vectors used to define the linear operator,
[001]=[102][010][111].
Multiplying both sides by our linear operator M we get,
M[001]=M[102]M[010]M[111].
Note that we know what M does to the vectors on the right so we can just substitute those values in and add,
M[001]=[101][010][111]=[020].
The resulting vector is the third column of our matrix,
M=[  0  2  0].
A similar process will yield the other collumns.
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