Question

Prove (as asserted in Sec. 9.6) that \(B A\) is linear if \(A\) and \(B\) are linear transformations. Prove also that \(A^{-1}\) is linear and invertible.

Short answer

Answer: If A and B are linear transformations and A is invertible, then their product BA is also a linear transformation, and the inverse A^{-1} is both linear and invertible.

Step-by-step solution

Proving BA is linear

To prove that the product of two linear transformations is also a linear transformation, we need to show that BA satisfies the two conditions for linearity: a) \(B A(u+v) = B A(u) + B A(v)\) for all vectors u, v. b) \(B A(c u) = c B A(u)\) for all vectors u and scalars c. Let A and B be linear transformations, and let u, v be any vectors and c be any scalar. Then: 1. \(B A(u+v) = B (A(u+v)) = B (A(u) + A(v))\) (since A is linear) 2. \(B (A(u) + A(v)) = B A(u) + B A(v)\) (since B is linear) Thus, condition (a) is satisfied. Similarly, for condition (b): 1. \(B A(c u) = B (A(c u)) = B (c A(u))\) (since A is linear) 2. \(B (c A(u)) = c B A(u)\) (since B is linear) Hence, condition (b) is satisfied as well. Therefore, BA is also a linear transformation.

Proving A^{-1} is linear

We will now prove that the inverse of a linear transformation A, denoted by A^{-1}, is also a linear transformation. To do so, we must show that A^{-1} satisfies the two conditions for linearity: a) \(A^{-1}(u+v) = A^{-1}(u) + A^{-1}(v)\) for all vectors u, v. b) \(A^{-1}(c u) = c A^{-1}(u)\) for all vectors u and scalars c. Let A be a linear transformation, and let u, v be any vectors and c be any scalar. Then, applying A to both sides of the conditions: 1. A(A^{-1}(u+v)) = A(A^{-1}(u) + A^{-1}(v)) 2. A(A^{-1}(c u)) = A(c A^{-1}(u)) Using the property that \(AA^{-1} = I\), the identity matrix, we have: 1. \(u+v = A^{-1}(u) + A^{-1}(v)\) (since A is linear) 2. \(c u = c A^{-1}(u)\) (since A is linear) By showing that A^{-1} satisfies the conditions for linearity, we have proven that A^{-1} is also a linear transformation.

Proving A^{-1} is invertible

Finally, we need to show that A^{-1} is invertible. To do this, we must show that there exists a matrix that when multiplied with A^{-1}, produces the identity matrix I. Since we know A is invertible, by the definition of an inverse, AA^{-1} = I, and A^{-1}A = I. Therefore, we have found the matrix A, which when multiplied with A^{-1} produces the identity matrix I. Thus, A^{-1} is invertible. In conclusion, we have proven that if A and B are linear transformations: 1. Their product BA is also a linear transformation. 2. The inverse of A, A^{-1}, is both linear and invertible.

Key concepts

Linearity of Matrix Product

When we talk about linear transformations in the realm of mathematics, particularly in linear algebra, we're looking at functions that map vectors to other vectors in a way that preserves the operations of vector addition and scalar multiplication. Understanding whether the matrix product of two linear transformations results in another linear transformation is crucial for constructing more complex systems.

Based on the steps detailed in the exercise solution, we can see that if we have two matrices, A and B, representing linear transformations, their product, denoted as BA, will also represent a linear transformation. This is evident from the linearity properties shown where the sum and scalar multiple of vectors are preserved after applying the product transformation.

• Sum preservation: If you add two vectors and then transform them, it's the same as if you transformed each separately and then added them.
• Scalar multiplication preservation: If you scale a vector and then transform it, it's equivalent to transforming the vector first and then scaling the result.

This outcome is fundamental to the concepts of linear algebra, as it ensures that the system's linearity is maintained even when transformations are compounded. Thus, in simpler terms, combining linear functions won't surprisingly produce non-linear results.

Inverse of Linear Transformation

Imagine undoing something perfectly—like perfectly unmixing two paints. In linear transformations, the inverse function 'un-does' whatever the original function did. When you have a linear transformation represented by a square matrix A, the inverse transformation, denoted as A^-1, reverses A's effects.

To prove that the inverse of a linear transformation is linear, we look to the original solution steps. It was demonstrated that applying A^-1 to both u+v and cu and then using the properties of the identity matrix, which is essentially the 'do nothing' transformation, reinforces that A^-1 meets the necessary linearity conditions—another key checkbox in the list of linear transformation properties.

In practical terms, if you've transformed a space with A, applying A^-1 brings everything back to where it started, and this 'reverse journey' is predictable and linear, much like the forward one. This is why finding inverses is powerful; it's the key to understanding systems that can be reversed and controlled.

Properties of Linear Transformations

Linear transformations are the backbone of linear algebra, acting like a set of rules that ensure transformations are predictable and stable. These properties include preserving addition and scalar multiplication, which were touched upon in proving the linearity of matrix products and inverses.

But there's more to linear transformations than just these operations. Here's a quick rundown on some essential properties they exhibit:

• Additivity: Transforming a sum of vectors is equivalent to summing their individual transformations.
• Homogeneity of degree 1 (scalar multiplication): Scaling a vector and then transforming it is the same as transforming the vector and then scaling the transformation.
• One-to-one mapping: Each input vector has a distinct output vector, essential for transformation reversibility.
• Transforming a zero vector: A linear transformation will always map the zero vector to another zero vector in the target space.

These properties create an environment where transformations are not fraught with surprises, enabling mathematicians and scientists to build upon further operations with certainty.

Identity Matrix

In the world of matrix transformations, the identity matrix plays the role of the numerical one in multiplication. The identity matrix, denoted by I, is special because multiplying any matrix or vector by I results in the original matrix or vector—no change occurs.

It's characterized by 1's down the main diagonal and 0's everywhere else. This unique pattern ensures that any value it interacts with remains untouched. The identity matrix is essential when discussing inverses of linear transformations—just as the number one is the multiplicative identity for real numbers, so is the identity matrix for matrix multiplication.

When you confirm that A^-1A = I and AA^-1 = I in the exercise, it's similar to proving that a number multiplied by its reciprocal equals one. This not only showcases the reversibility of a linear transformation but solidifies our understanding of the stability at play within these mathematical systems.