Linear Algebra and Its Applications, Exercise 2.6.13

Exercise 2.6.13. Suppose that A is a linear transformation from the xy plane to itself. If a transformation A^{-1} exists such that A^{-1}(Ax) = A(A^{-1}x) = x show that A^{-1} is also linear. Also show that if M is the matrix representing A then the matrix representing A^{-1} must be M^{-1}.

Answer: Consider the expression cA^{-1}x + dA^{-1}y for any x and y. We know from the definition of A^{-1} that

cA^{-1}x + dA^{-1}y = A^{-1}[A(cA^{-1}x + dA^{-1}y)]

Since A is a linear transformation we then have

A^{-1}[A(cA^{-1}x + dA^{-1}y)] = A^{-1}[cA(A^{-1}x) + dA(A^{-1}y)]

and using the definition of A^{-1} we have A(A^{-1}x) = x and A(A^{-1}y) = y so that

A^{-1}[cA(A^{-1}x) + dA(A^{-1}y)] = A^{-1}(cx + dy)

From our final result and initial expression we have

A^{-1}(cx + dy) = cA^{-1}x + dA^{-1}y

for all x and y. So if A is a linear transformation and A^{-1} exists then A^{-1} is a linear transformation also.

Now suppose that M is the matrix representing A and N is the matrix representing A^{-1}. By 2V on page 123 the composition of A and A^{-1} is itself a linear transformation, with AA^{-1} represented by the product matrix MN and A^{-1}A represented by the product matrix NM.

But since  A^{-1}(Ax) = A(A^{-1}x) = x for all x we know that the compositions of A and A^{-1} are equivalent to the identity transformation represented by the identity matrix I. We therefore have MN = I and NM = I which implies that N = M^{-1}.

So if M is the matrix representing A then M^{-1} is the matrix representing A^{-1}.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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2 Responses to Linear Algebra and Its Applications, Exercise 2.6.13

  1. Stacy says:

    hey u free i wanna ask for help with a problem if you wanna.
    Nelly’s shadow is 1.5m a tree that is 30 meters tall has a shadow of 25m how tall is Nelly?

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