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Math Help - Consider this matrix and prove...

  1. #1
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    Consider this matrix and prove...

    Consider the n x n matrix with all zeros except for the real numbers α1 ... αn on the diagonal. Prove, by checking that it preserve addition and scalar multiplication, that this matrix corresponds to a linear transformation of R^n

    That a1 and an are alpha-1 and alpha-n.

    I really hope this question makes sense to someone. I'm not the strongest at math and I need this course to get into a Master's program. I really hope you can help me!

    Thanks so much!
    - H.

    Edit: I posted this in another form area by accident, and I hope it's ok if I reposted it here. :/
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  2. #2
    Grand Panjandrum
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    Re: Consider this matrix and prove...

    Quote Originally Posted by hyrule View Post
    Consider the n x n matrix with all zeros except for the real numbers α1 ... αn on the diagonal. Prove, by checking that it preserve addition and scalar multiplication, that this matrix corresponds to a linear transformation of R^n

    That a1 and an are alpha-1 and alpha-n.

    I really hope this question makes sense to someone. I'm not the strongest at math and I need this course to get into a Master's program. I really hope you can help me!

    Thanks so much!
    - H.

    Edit: I posted this in another form area by accident, and I hope it's ok if I reposted it here. :/
    That an operator H is a linear transformation on \mathbb{R}^n means:

    H(\alpha x+\beta y)=\alpha Hx +\beta Hy

    for any x and y in \mathbb{R}^n and \alpha and \beta in \mathbb{R}

    Here Hy means your matrix is H , y is a (column) vector in \mathbb{R}^n and Hy denote the matrix prioduct of H and x.

    Now what have you tried, and what problems are you having?

    CB
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  3. #3
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    Re: Consider this matrix and prove...

    I haven't tried too much with this question because I am just stumped at this entire question. I know about transformation matrices and all that, but I wondered if I was supposed to multiply the matrix n x n by x1, x2, x3? I'm just not very strong in this type of math, heh.
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  4. #4
    Grand Panjandrum
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    Re: Consider this matrix and prove...

    Quote Originally Posted by hyrule View Post
    I haven't tried too much with this question because I am just stumped at this entire question. I know about transformation matrices and all that, but I wondered if I was supposed to multiply the matrix n x n by x1, x2, x3? I'm just not very strong in this type of math, heh.
    Because your matrix is diagonal

    Hx=\left[ \begin{array}{cccc}a_1 & 0 &...& 0 \\ 0 & a_2 & ... & 0 \\ \vdots & &\ddots & \vdots \\ 0&...&0&a_n \end{array} \right] \left[ \begin{array}{c} x_1\\x_2 \\ \vdots \\ x_n \end{array} \right]=\left[ \begin{array}{c} a_1 x_1\\a_2 x_2 \\ \vdots \\ a_n x_n \end{array} \right]

    CB
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  5. #5
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    Re: Consider this matrix and prove...

    Oooh ok, I get it. So because I have alpha-n number of all real numbers in the diagonal matrix, I needed to multiply it with Xn number of variables.

    I had a suspiscian that I would have to multiply by x variables, but I just wasn't confident in my own answer. You've really helped me understand it better, thank you!!
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