Problem 41 - Differential Equations and Linear Algebra

Question

If \(T: V \rightarrow W\) is an invertible linear transformation (that is, \(T^{-1}\) exists), show that \(T^{-1}: W \rightarrow V\) is also a linear transformation.

Step-by-Step Solution

Answer

To show that if \(T: V \rightarrow W\) is an invertible linear transformation, its inverse \(T^{-1}: W \rightarrow V\) is also a linear transformation. First, for additivity: if we consider two vectors \(u, v \in W\), since \(T\) is invertible, we can say that \(u = T(x)\) and \(v = T(y)\) for some \(x, y \in V\). Now, \(u+v = T(x+y)\) and applying \(T^{-1}\) we get \(T^{-1}(u+v) = x+y = T^{-1}(u) + T^{-1}(v)\). So, additivity holds. Secondly, for homogeneity: considering \(u \in W\) and scalar \(c\), since \(T\) is invertible, \(u = T(x)\) for some \(x \in V\). Then \(cu = T(cx)\), and applying \(T^{-1}\) we get \(T^{-1}(cu) = cx = cT^{-1}(u)\). So, homogeneity holds. Hence, \(T^{-1}\) satisfies additivity and homogeneity properties, it is a linear transformation. Therefore, we have shown that the inverse of an invertible linear transformation is also a linear transformation.

Step 1: Recall the definitions of additivity and homogeneity

To be a linear transformation, \(T^{-1}\) must satisfy these two conditions: 1. Additivity: \(T^{-1}(u + v) = T^{-1}(u) + T^{-1}(v)\) for all \(u, v \in W\). 2. Homogeneity: \(T^{-1}(cu) = cT^{-1}(u)\) for all \(u \in W\) and scalar \(c\). We need to verify that both of these conditions hold for the inverse transformation, \(T^{-1}\).

Step 2: Show Additivity for \(T^{-1}\)

Consider two vectors \(u, v \in W\). Since \(T\) is an invertible linear transformation, \(u = T(x)\) and \(v = T(y)\) for some \(x, y \in V\). Now, let's consider the sum of \(u\) and \(v\): \(u + v = T(x) + T(y)\). Since \(T\) is a linear transformation, it follows that \(T(x+y) = T(x) + T(y)\) by additivity. Therefore, \(u + v = T(x+y)\). Now, to find the inverse transformation of the sum \(u+v\), we apply \(T^{-1}\) to both sides: \(T^{-1}(u+v) = T^{-1}(T(x+y))\). By definition of the inverse, \(T^{-1}(T(x+y)) = x+y\). Also, \(T^{-1}(u) = x\) and \(T^{-1}(v) = y\). Therefore, \(T^{-1}(u+v) = x+y = T^{-1}(u) + T^{-1}(v)\). Thus, additivity holds for \(T^{-1}\).

Step 3: Show Homogeneity for \(T^{-1}\)

Consider a vector \(u \in W\) and a scalar \(c\). Since \(T\) is an invertible linear transformation, \(u = T(x)\) for some \(x \in V\). Now, let's consider the result of scaling \(u\) by \(c\): \(cu = cT(x)\). Since \(T\) is a linear transformation, it follows that \(T(cx) = cT(x)\) by homogeneity. Therefore, \(cu = T(cx)\). Now, to find the inverse transformation of the scaled vector \(cu\), we apply \(T^{-1}\) to both sides: \(T^{-1}(cu) = T^{-1}(T(cx))\). By definition of the inverse, \(T^{-1}(T(cx)) = cx\). Also, \(T^{-1}(u) = x\). Therefore, \(T^{-1}(cu) = cx = cT^{-1}(u)\). Thus, homogeneity holds for \(T^{-1}\).

Step 4: Conclusion

Since \(T^{-1}\) satisfies the properties of additivity and homogeneity, it is a linear transformation. Therefore, we have shown that if \(T: V \rightarrow W\) is an invertible linear transformation, its inverse \(T^{-1}: W \rightarrow V\) is also a linear transformation.

Key Concepts

Additivity of Linear Transformations

The concept of additivity is foundational for understanding linear transformations. In the context of linear algebra, a transformation is said to be additive if it preserves vector addition. This means that when you take any two vectors in the domain of the transformation and add them together, after applying the transformation, the result is the same as if you applied the transformation to each vector individually and then added the results.
For example, consider a linear transformation denoted by the function notation, \( T \. With additivity, we can say that \( T(u + v) = T(u) + T(v) \), for any vectors \( u \) and \( v \). Additivity is crucial for the structure of vector spaces because it preserves the vector space operations when moving between different spaces via the transformation.
In the step by step solution provided, additivity is demonstrated for the inverse \( T^{-1} \), ensuring that the inverse function is also a linear transformation. We see this when \( T^{-1}(u+v) = T^{-1}(u) + T^{-1}(v) \), thus fulfilling a core requirement for linearity.

Homogeneity of Linear Transformations

Homogeneity is another essential property of linear transformations. It comes into play when we consider scalar multiplication, a vital operation in vector spaces. A transformation is homogeneous if, when you multiply a vector by a scalar and then apply the transformation, it's equivalent to applying the transformation first and multiplying the result by that scalar.
Mathematically, we express this idea as \( T(cv) = cT(v) \) for any vector \( v \) and scalar \( c \). In simple terms, the transformation distributes over scalar multiplication, which is a fundamental part of preserving the structure of vector spaces.
In the provided solution, homogeneity is verified for the inverse \( T^{-1} \) by showing that \( T^{-1}(cu) = cT^{-1}(u) \). Confirming homogeneity for \( T^{-1} \) ensures that scalar multiplication behaves consistently in both the original and inverted transformations, maintaining the linear nature of the inverse.

Properties of Linear Transformations

Linear transformations have a set of properties that make them such a powerful tool in linear algebra. These include additivity and homogeneity, as discussed earlier, but also extend to concepts like the preservation of the zero vector and the ability to transform lines into lines without curving them.
A linear transformation is essentially a function that moves vectors from one vector space to another in a way that respects both vector addition and scalar multiplication. These linear properties ensure that the structure of the vector space is maintained after transformation.
When we speak about an invertible linear transformation, it means there exists an inverse function that can reverse the transformation effects. If the original transformation is linear, so must be the inverse one, as demonstrated in the step by step solution, for the preserved properties like additivity and homogeneity ensure the functionality and applications of linear concepts in diverse mathematical and applied contexts.

Vector Spaces

Vector spaces are fundamental structures in linear algebra. A vector space is essentially a collection of objects that we can add together and multiply by scalars to produce new objects within the same collection. These objects are called vectors, and the rules for addition and scalar multiplication are dictated by a set of axioms that all vector spaces must follow.
Structural properties such as associativity, commutativity of addition, and distribution of scalar multiplication over vector addition are vital. These properties ensure that vectors form a well-defined mathematical space where linear transformations can operate.
For a linear transformation to be considered invertible, as in the exercise provided, there must be a two-way street of operation between two vector spaces. The existence of an invertible transformation implies a kind of symmetry in the operations and structure between the vector spaces, reflecting one another's properties through the bijective, or one-to-one and onto, nature of the invertible transformation.