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    Linear_notes

    Uploaded by

    Rik Ledger

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    © All Rights Reserved
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    Linear_notes

    Uploaded by

    Rik Ledger
    2 views1 page

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    © © All Rights Reserved

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    2 views1 page

    Linear_notes

    Uploaded by

    Rik Ledger

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    Theorem 1 (Rank-Nullity Theorem).

    Let V be a finite-dimensional vector


    space, and let T : V → W be a linear transformation, where W is also a
    finite-dimensional vector space. Then

    dim(ker(T )) + dim(Im(T )) = dim(V ).

    Theorem 2 (Cayley–Hamilton Theorem). Let A be a square n × n matrix


    with characteristic polynomial χA (λ) = det(λI − A). Then A satisfies its
    own characteristic equation; that is,

    χA (A) = 0.

    Theorem 3 (Diagonalization Criterion). A square n × n matrix A (over


    a field F) is diagonalizable over F if and only if there exists a basis of Fn
    consisting of eigenvectors of A. Equivalently, A can be written as

    A = P DP −1 ,

    where D is a diagonal matrix and P is invertible.

    Theorem 4 (Jordan Canonical Form). If A is a square matrix over an


    algebraically closed field F (e.g., C), then A is similar to a block-diagonal
    matrix (the Jordan form)
     
    Jn1 (λ1 ) 0
    A∼J =
     .. ,

    .
    0 Jnk (λk )

    where each Jni (λi ) is a Jordan block corresponding to an eigenvalue λi . This


    form is unique up to the order of the blocks.

    Theorem 5 (Spectral Theorem for Symmetric Matrices). Let A be a real


    symmetric n × n matrix. Then A can be orthogonally diagonalized; that is,
    there exists an orthogonal matrix Q (so QT Q = I) and a diagonal matrix D
    such that
    A = QDQT ,
    where D contains the real eigenvalues of A on its diagonal.

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