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Math Help - linear transformations

  1. #1
    Junior Member
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    linear transformations

    I need help with this problem:

    1. let V be vector space with a finite dimension and let T:V to V be linear transformation. prove that if T is not isomorphism, so there is a basis B of V that [T]B is a matrix with a zero column.

    2. Prove that if A is singular so it's similar to a matrix with a zero column.

    Thanks ahead
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  2. #2
    MHF Contributor

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    Quote Originally Posted by omert View Post
    I need help with this problem:

    1. let V be vector space with a finite dimension and let T:V to V be linear transformation. prove that if T is not isomorphism, so there is a basis B of V that [T]B is a matrix with a zero column.

    2. Prove that if A is singular so it's similar to a matrix with a zero column.

    Thanks ahead
    part 2 is just rephrasing part 1 of your question in terms of matrices. for part 1, since T is not an isomorphism, \ker T \neq (0). thus \dim \ker T \geq 1. choose a basis for \ker T, say B_1=\{v_1, \cdots , v_k \},

    and extend it to a basis B for V. then T(v_1)=\bold{0} is the first column of [T]_B. \ \Box
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