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# 数学代写|拓扑学代写TOPOLOGY代考|TMA4190 LINEAR TRANSFORMATIONS

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## 数学代写|拓扑学代写TOPOLOGY代考|LINEAR TRANSFORMATIONS

Let $L$ and $L^{\prime}$ be linear spaces with the same system of scalars. A mapping $T$ of $L$ into $L^{\prime}$ is called a linear transformation if
$$T(x+y)=T(x)+T(y) \quad \text { and } \quad T(\alpha x)=\alpha T(x),$$
or equivalently, if
$$T(\alpha x+\beta y)=\alpha T(x)+\beta T(y) .$$
A linear transformation of one linear space into another is thus a homomorphism of the first space into the second, for it is a mapping which preserves the linear operations. $T$ also preserves the origin and negatives, for $T(0)=T(0 \cdot 0)=0 \cdot T(0)=0$ and
$$T(-x)=T((-1) x)=(-1) T(x)=-T(x) .$$
The importance of linear spaces lies mainly in the linear transformations they carry, for vast tracts of algebra and analysis, when placed in their proper context, reduce to the study of linear transformations of one linear space into another. The theory of matrices, for instance, is one small corner of this subject, as are the theory of certain types of differential and integral equations and the theory of integration in its most elegant modern form.

In the following examples we leave it to the reader to show that each mapping described actually is a linear transformation.

## 数学代写|拓扑学代写TOPOLOGY代考|ALGEBRAS

A linear space $A$ is called an algebra (see Sec. 20) if its vectors can be multiplied in such a way that $A$ is also a ring in which scalar multiplication is related to multiplication by the following property:
$$\alpha(x y)=(\alpha x) y=x(\alpha y) .$$
The concept of an algebra is therefore a natural combination of the concepts of a linear space and a ring. Figure 34 illustrates the manner in which the major algebraic systems defined in this chapter are related to one another.

Since an algebra is a linear space, all the ideas developed in Secs. 42 and 43 are immediately applicable. Some algebras are real and some are complex, and every algebra has a well-defined dimension. Furthermore, since an algebra is also a ring, it may be commutative or non-commutative, and may or may not have an identity; and if it does have an identity, then we can speak of its regular and singular elements. A division algebra is an algebra with identity which, as a ring, is a division ring. A subalgebra of an algebra $A$ is a non-empty subset $A_{0}$ of $A$ which is an algebra in its own right with respect to the operations in $A$. This condition evidently means that $A_{0}$ is closed under addition, scalar multiplication, and multiplication.

## 数学代写|拓扑学代写TOPOLOGY代考|LINEAR TRANSFORMATIONS

$$T(x+y)=T(x)+T(y) \quad \text { and } \quad T(\alpha x)=\alpha T(x),$$

$$T(\alpha x+\beta y)=\alpha T(x)+\beta T(y) .$$

$$T(-x)=T((-1) x)=(-1) T(x)=-T(x) .$$

## 数学代写|拓扑学代写TOPOLOGY代考|ALGEBRAS

$$\alpha(x y)=(\alpha x) y=x(\alpha y) .$$

## MATLAB代写

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