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Linear Algebra 2.3

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Section 1

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T is invertible iff A

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Algebra Math

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Mar 1, 2020

Cards (9)

Section 1

(9 cards)

T is invertible iff A

Front

is an invertible matrix

Back

When is linear transformation T: R^n -> R^n invertible?

Front

If there are functions S: R^n -> R^n such that 1) S(T(x)) = x for all x in R^n and 2) T(S(x)) = x for all x in R^n.

Back

singular matrices

Front

noninvertible

Back

nonsingular matrices

Front

invertible

Back

Invertible Matrix Theorem (square matrix)

Front

1) A is an invetible matrix. 2) A is row equivalent to the nxn identity matrix. 3) A has n pivot positions. 4) The equation Ax=0 has only the trivial solution. 5) The columns of A form a linearly independent set. 6) The linear transformation x I-> Ax is one-to-one. 7) The equation Ax=b has at least one solution for each b in R^n. 8) The columns of A span R^n. 9) The linear transformation x I-> Ax maps R^n onto R^n. 10) There is an nxn matrix C such that CA=I. 11) There is an nxn matrix D such that AD=I. 12) A^T is an invertible matrix.

Back

What does "The equation Ax = b has a unique solution for each b in R^n" imply?

Front

1) A is row equivalent to the identity matrix and 2) A is invertible

Back

If A and B are square matrices and AB = I

Front

Then A and B are both invertible with B=A^-1 and A=B^-1

Back

How would you write S as the Inverse of T?

Front

S = T^-1

Back

S(x) = A^(-1)x

Front

unique function

Back