Linear Algebra/Linear Transformations – Wikibooks, open books for an open world
A linear transformation is an important concept in mathematics because many real world phenomena can be approximated by linear models. Another example instead of rotating a vector, we stretch it, so a vector becomes , for example. becomes Or, if we look at the projection of one vector onto the x axis – extracting its x component – , e.g. from we get These examples are all an example of a mapping between two vectors, and are all linear transformations. If the rule transforming the matrix is called , we often write for the mapping of the vector by the rule . is often called the transformation. Note we do not always write brackets like when we write functions. However we should write brackets, especially when we want to express the mapping of the sum or the product or the combination of many vectors. Suppose one has a field K, and let x be an element of that field. Let O be a function taking values from K where O(x) is an element of a field J. Define O to be a linear form if and only if: Suppose one has a vector space V, and let x be an element of that vector space. Let F be a function taking values from V where F(x) is an element of a field K. Define F to be a linear form if and only if: This time, instead of a field, let us consider functions from one vector space into another vector space. Let T be a function taking values from one vector space V where L(V) are elements of another vector space. Define L to be a linear transformation when it: Note that not all transformations are linear. Many simple transformations that are in the real world are also non-linear. Their study is more difficult, and will not be done here. For example, the transformation S (whose input and output are both vectors in R2) defined by This means that T, whatever transformation it may be, maps vectors in the vector space V to a vector in the vector space W. Here are some examples of some linear transformations. At the same time, let’s look at how we can prove that a transformation we may find is linear or not. Let us take the projection of vectors in R2 to vectors on the x-axis. Let’s call this transformation T. This is the same vector as above, so under the transformation T, scalar multiplication is preserved. When we want to disprove linearity – that is, to prove that a transformation is not linear, we need only find one counter-example. If we can find just one case in which the transformation does not preserve addition, scalar multiplication, or the zero vector, we can conclude that the transformation is not linear. Given the above, determine whether the following transformations are in fact linear or not. Write down each transformation in the form T:V ->, W, and identify V and W. (Answers follow to even-numbered questions): We have some fundamental concepts underlying linear transformations, such as the kernel and the image of a linear transformation, which are analogous to the zeros and range of a function. The kernel of a linear transformation T: V ->, W is the set of all vectors in V which are mapped to the zero vector in W, ie., The kernel of a transform T: V->,W is always a subspace of V. The dimension of a transform or a matrix is called the nullity.. The image of a linear transformation T:V->,W is the set of all vectors in W which were mapped from vectors in V. For example with the trivial mapping T:V->,W such that Tx=0, the image would be 0. (What would the kernel be?). Source.