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Math Help - Standard Matrices and basis vectors

  1. #1
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    Standard Matrices and basis vectors

    I am taking an online course for linear algebra and getting a response from my teacher takes a while. Could you guys please help me understand parts a b , and d? I'm not really looking for the answer so much as the steps behind it. Thank you in advance. Standard Matrices and basis vectors-6.jpg
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  2. #2
    A Plied Mathematician
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    Can you find a matrix representation of T? I think that would get you started.
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  3. #3
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    Quote Originally Posted by Ackbeet View Post
    Can you find a matrix representation of T? I think that would get you started.

    3 -1
    -1 1
    0 5

    I think...but what about e sub1 and e sub 2 that's what's really tripping me up.
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  4. #4
    A Plied Mathematician
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    I agree with your representation of T. Now, e_{i} is a standard notation for basis vectors. I would guess that

    e_{1}=\left[\begin{matrix}1\\ 0\end{matrix}\right], and

    e_{2}=\left[\begin{matrix}0\\ 1\end{matrix}\right].

    So, what do you suppose T(e_{1}) is?
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  5. #5
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    The part I don't understand. Would it be a 2x2 or 3x3? I would assume it comes out to be  \begin{bmatrix}<br />
1\, \, \, 0 & \\ <br />
0\, \, \,  1 & <br />
\end{bmatrix}
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  6. #6
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    Quote Originally Posted by Jukodan View Post
    The part I don't understand. Would it be a 2x2 or 3x3? I would assume it comes out to be  \begin{bmatrix}<br />
1\, \, \, 0 & \\ <br />
0\, \, \,  1 & <br />
\end{bmatrix}
    Neither. And, no, the matrix form is not \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}.

    You are told that the linear transfromation is "from R^2 to R^3" so you will be multiplying a matrix by a "column matrix" with 2 entries (actually 1 column, 2 rows as a matrix) so there must be two columns in you matrix to be able to multiply. The result is in [itex]R^3[/itex] so it will be a "column matrix" with 3 entries (1 column, 3 rows) which means your matrix must have 3 rows to give 3 results. The matrix representing this linear transformation must be "2 by 3" (2 columns, 3 rows).

    Do as Ackbeet suggested: since you titled this "Standard matrices and basis vectors" he, and I, assume that e_1= \begin{bmatrix}1 \\ 0 \end{bmatrix} and e_2= \begin{bmatrix}0 \\ 1\end{bmatrix}.

    You are told that T(x_1, y_1)= (3x_1- x_2, -x_1+ x_2, 5x_2) so what is T(1, 0)? What is T(0, 1)?

    Now use the fact that \begin{bmatrix}a & b\\ c & d \\ e & f\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}= \begin{bmatrix}a \\ c \\ e\end{bmatrix} and \begin{bmatrix}a & b\\ c & d \\ e & f\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix}= \begin{bmatrix}b \\ d \\ f\end{bmatrix} to find a, b, c, d, e, and f.
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  7. #7
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    So then T(e_1)=\begin{bmatrix}<br />
3\\ <br />
-1\\ <br />
0<br />
\end{bmatrix} and T(e_2)=\begin{bmatrix}<br />
-1\\ <br />
1\\ <br />
5<br />
\end{bmatrix}

    Am I following you correctly?
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  8. #8
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    Yes, that is correct. And, so now what are a, b, c, d, e, f? In other words, what is the matrix representing this linear transformation in the standard basis?
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  9. #9
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    The standard matrix is ..... \begin{bmatrix}<br />
3\:-1 & \\ <br />
-1\;\; 1& \\ <br />
 0\;\;5& <br />
\end{bmatrix}
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  10. #10
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    Quote Originally Posted by Jukodan View Post
    The standard matrix is ..... \begin{bmatrix}<br />
3\:-1 & \\ <br />
-1\;\; 1& \\ <br />
 0\;\;5& <br />
\end{bmatrix}
    Yes, that is correct- very good!
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  11. #11
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    I thank both of you very much for guiding me and shedding light on the problem. I understand this a lot better now. Both of you were very helpful, thank you again.
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