Recent content by Ioiô

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    Trace(matrix) = 0 and the dimension of subspace

    Homework Statement Claim: Matrices of trace zero for a subspace of M_n (F) of dimension n^2 -1 where M_n (F) is the set of all nxn matrices over some field F. Homework Equations Tr(M_n) = sum of diagonal elements The Attempt at a Solution I view the trace Tr as a linear...
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    Graduate Triangular Form for Linear Transformations: Finding Basis for Null Spaces

    In (T+1) w_2 \in S(w_1) S means subspace. (I know it is confusing when S is also used for the matrix relating the two basis.) The space V is a direct sum of V1 = null(T+1)^2 and V2 = null(T-2). Is this the reason for the 2x2 matrix block, the diagonals are all -1 and the 1x1 matrix block...
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    Graduate Triangular Form for Linear Transformations: Finding Basis for Null Spaces

    I am confused on how to find a matrix B in triangular form for some linear transformation T over a basis \{v_1,v_2, v_3\} . Suppose we are given a minimal polynomial m(x) = (x+1)^2 (x-2). Do I want to find a basis \{w_1,w_2\} for null(T+1)^2 such that (T+1) w_1 = 0 and (T+1) w_2 \in...