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# 1) Define logical equivalence. 2) What does the statement: The n x n matrix A is invertible. 3) How do I justify these statements:

1) Define logical equivalence.
2) What does the statement: The n x n matrix A is invertible.
3) How do I justify these statements: a) A is an invertible matrix b) A is row equivalent to the n x n identity matrix c) A has n pivot positions d) The equation Ax =0 has only the trivial solution. e) The equation Ax = b has at least one solution for each b in R
f) The columns of A span R^n g) The linear transformation x (arrow) Ax maps R^n onto R^n h) There is an n x n matrix D such that AD =I j) The columns of A form a basis of R^n
are equivalent to to the statement the n x n matrix A is invertable

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