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Math Help - Matrix of a linear transformation

  1. #1
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    Lightbulb Matrix of a linear transformation

    Consider the following problem.

    Let M denote the set of 2x2 real matrices. Let A be an element of M with trace 2 and determinant -3. Identifying M with R4, consider the linear transformation T: M -> M defined by T(B) = AB. Then which of the following statements are true?

    a) T is diagonalizable. b) 2 is an eigenvalue of T c) T is invertible d) T(B) = B for some non-zero matrix B in the set M.

    Based on the opinion that the matrix of a linear transformation is the matrix which is multiplied by the input matrix of the domain, to get the output matrix belonging to the co-domain.

    Here the definition of the transformation gives us the impression that it is obvious that the matrix A is the matrix of the linear transformation. So as per the given information since determinant of A is not zero it is invertible, which means it is also diagonalizable. Therefore options a and c are true. Since the trace is 2 and the determinant is -3, the two eigen values are -3 and +1 which shows that option b is not true. Similarly, since A cannot be the identity matrix, T(B) = AB can never be B itself. This means option d is also not true.

    My question is this.

    What does the phrase, "Identifying M with R4" have to do with the problem?

    Is there something overlooked by me due to my ignoring of the significance of this phrase?
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  2. #2
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    Re: Matrix of a linear transformation

    "Identifying M with R^4" means thinking of \begin{bmatrix}a & b \\ c & d \end{bmatrix} as being the same as (a, b, c, d) with the "usual" addition and scalar multiplication. That effectively means \begin{bmatrix}a & b \\ c & d \end{bmatrix}+ \begin{bmatrix}e & f \\ g & h\end{bmatrix}= \begin{bmatrix}a+ e & b+ f \\ c+ g & d+ h\end{bmatrix} and \alpha\begin{bmatrix}a & b \\ c & d \end{bmatrix}= \begin{bmatrix}\alpha a & \alpha b \\ \alpha c & \alpha d \end{bmatrix}.
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  3. #3
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    Re: Matrix of a linear transformation

    Thank you HallsofIvy... I sort of assumed it to be a default info... do you find any other mistake with the conclusions i've arrived at the options of the givenquestion?
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