数学代写|代数学代写Algebra代考|МATH611

2022年9月30日

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数学代写|代数学代写Algebra代考|Rotations in Higher Dimensions

While projections and reflections work regardless of the dimension of the space that they act on, extending rotations to dimensions higher than 2 is somewhat delicate. For example, in $\mathbb{R}^3$ it does not make sense to say something like “rotate $\mathbf{v}=(1,2,3)$ counter-clockwise by an angle of $\pi / 4 “$ “-this instruction is under-specified, since one angle alone does not tell us in which direction we should rotate.

In $\mathbb{R}^3$, we can get around this problem by specifying an axis of rotation via a unit vector (much like we used a unit vector to specify which line to project onto or reflect through earlier). However, we still have the problem that a rotation that looks clockwise from one side looks counter-clockwise from the other. Furthermore, there is a slightly simpler method that extends more straightforwardly to even higher dimensions-repeatedly rotate in a plane containing two of the coordinate axes.

To illustrate how to do this, consider the linear transformation $R_{y z}^\theta$ that rotates $\mathbb{R}^3$ by an angle of $\theta$ around the $x$-axis, from the positive $y$-axis toward the positive $z$-axis. It is straightforward to see that rotating $\mathbf{e}1$ in this way has no effect at all (i.e., $R{y z}^\theta\left(\mathbf{e}1\right)=\mathbf{e}_1$ ), as shown in Figure 1.20. Similarly, to rotate $\mathbf{e}_2$ or $\mathbf{e}_3$ we can just treat the $y z$-plane as if it were $\mathbb{R}^2$ and repeat the derivation from Figure $1.18$ to see that $R{y z}^\theta\left(\mathbf{e}2\right)=(0, \cos (\theta), \sin (\theta))$ and $R{y z}^\theta\left(\mathbf{e}_3\right)=(0,-\sin (\theta), \cos (\theta))$

数学代写|代数学代写Algebra代考|Composition of Linear Transformations

As we mentioned earlier, there are some simple ways of combining linear transformations to create new ones. In particular, we noted that if $S, T: \mathbb{R}^n \rightarrow$ $\mathbb{R}^m$ are linear transformations and $c \in \mathbb{R}$ then $S+c T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is also a linear transformation.

We now introduce another method of combining linear transformations that is slightly more exotic, and has the interpretation of applying one linear transformation after another:

Suppose $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $S: \mathbb{R}^m \rightarrow \mathbb{R}^p$ are linear transformations. Then their composition is the function $S \circ T: \mathbb{R}^n \rightarrow \mathbb{R}^p$ defined by
$$(S \circ T)(\mathbf{v})=S(T(\mathbf{v})) \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^n \text {. }$$
That is, the composition $S \circ T$ of two linear transformations is the function that has the same effect on vectors as first applying $T$ to them and then applying $S$. In other words, while $T$ sends $\mathbb{R}^n$ to $\mathbb{R}^m$ and $S$ sends $\mathbb{R}^m$ to $\mathbb{R}^p$, the composition $S \circ T$ skips the intermediate step and sends $\mathbb{R}^n$ directly to $\mathbb{R}^p$, as illustrated in Figure 1.21.

Even though $S$ and $T$ are linear transformations, at first it is not particularly obvious whether or not $S \circ T$ is also linear, and if so, what its standard matrix is. The following theorem shows that $S \circ T$ is indeed linear, and its standard matrix is simply the product of the standard matrices of $S$ and $T$. In fact, this theorem is the main reason that matrix multiplication was defined in the seemingly bizarre way that it was.

In other words, $S \circ T$ is a function that acts on $\mathbf{v}$ in the exact same way as matrix multiplication by the matrix $[S][T]$. It thus follows from Theorem 1.4.1 that $S \circ T$ is a linear transformation and its standard matrix is $[S][T]$, as claimed.
By applying linear transformations one after another like this, we can construct simple algebraic descriptions of complicated geometric transformations. For example, it might be difficult to visualize exactly what happens to $\mathbb{R}^2$ if we rotate it, reflect it, and then rotate it again, but this theorems tells us that we can unravel exactly what happens just by multiplying together the standard matrices of the three individual linear transformations.

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数学代写|代数学代写Algebra代考| Higher Dimensions旋转

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$$(S \circ T)(\mathbf{v})=S(T(\mathbf{v})) \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^n \text {. }$$

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