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This is so because we want to use positiveorientations for $\phi$ for all three rotation matrices. For $\mx{R}_x$, for example, imagine that you are looking down the negative$x$-axis, which means you will see the $y$- and $z$-axes positively. This is not the case, for the $x$- and $z$-axes, when lookingdown the negative $y$-axis for $\mx{R}_y$. Note that since $\cos(-\phi)=\cos\phi$ and $\sin(-\phi)=-\sin\phi$,rotating in the negative direction will flip the signs only on the $\sin\phi$-terms.

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Now, it is possible to use Theorem 6/10 to find out what the matrices look like that expresses these. However, an alternative way when dealing with is simply to imagine the vectors $(1,0)$ and $(0,1)$ expressed in $\{\hat{\vc{e}}_1, \hat{\vc{e}}_2\}$ and then figure out how to set up a matrix that transforms those vectors into $\{\vc{e}_1, \vc{e}_2\}$. It is rather simple as seen below.

In practice, one may collapse some of the steps above in order to further save space. Note that this method works for any size of a square matrix (https://srirejekifastboat.com/serial-code/?file=6597). Finally, if the system does not have a solution, then the matrix is not invertible. The reader may try to multiply $\mx{A}$ and $\mx{A}^{-1}$ and make sure that the result is the identity matrix, $\mx{I}$.

Similar to shearing in two dimensions (Definition 6/12), the first subscript of $\mx{H}$ refers to which coordinate is changed and the second subscript refers to the coordinate that is used to scale by $s$ and add to that first coordinate. The following combinations are also possible, $\mx{H}_{xz}(s)$, $\mx{H}_{yz}(s)$, and $\mx{H}_{zx}(s)$.

The hidden layer uses 500 nodes and the rectified linear activation function. A sigmoid activation function is used in the output layer in order to predict class values of 0 or 1.

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Each of the l1 and l2 regularizers takes a single hyperparameter that controls the amount that each activity contributes to the sum. The l1_l2 regularizer takes two hyperparameters, one for each of the l1 and l2 methods.

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The latter is probably the preferred usage of activation regularization as described in “Deep Sparse Rectifier Neural Networks” in order to allow the model to learn to take activations to a true zero value in conjunction with the rectified linear activation function. Nevertheless, the two possible uses of activation regularization may be explored in order to discover what works best for your specific model and dataset.

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The topmost slider can be used to give $\mx{A}$ a different appearance. Click/press Forward to continue to the next step. Remember to click/press Reset if you have want the original matrices restored.

Normally, when we see expressions such as the first row in (6/50) are used to that $\mx{B}=\mx{C}$. That is certainly possible, however, as we saw in (6/49), neither of the terms in a matrix-matrix multiplication needs to be the zero matrix, $\mx{O}$, in order for the product to be the zero matrix. In this case, it means that $\mx{B} -\mx{C}$ does not need to be the zero matrix (moved here).

### Definition 6.9: Matrix-Matrix Multiplication

This notation is also used in Example 6/11. The rules of (Theorem 5/2) can be applied still, since we have only changed the notation.

You can see that the red vector then becomes $\hid{(\cos(\pi/6),\sin(\pi/6)=(\sqrt{3}/2,0/5)}$. The parallelograms of the vector pair are also shown. In the case of a rotation matrix (https://srirejekifastboat.com/serial-code/?file=7702), these becomes squares with area 1/0. This will be explained further in \linkref{Chapter}{ch_dt}, where it will become clear that the area is related to the determinant of the respective matrix. Click/press Forward to continue to the next type of matrix. Remember to click/press Reset if you have moved the vectors and want the original matrices restored.

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As we saw in Theorem 6/14, $\mx{A}\mx{B}$ is if $\mx{A}$ and $\mx{B}$ are. In the following interactive illustration, we will visualize the matrix-matrix multiplication between two matrices.

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Assume we have guessed that the inverse to $(\mx{A}\mx{B})$ is $\invmx{B}\invmx{A}$ and that we then want to check that it is correct. We can now multiply $\mx{A}\mx{B}$ with $\invmx{B}\invmx{A}$ and see the result: $(\mx{A}\mx{B})(\invmx{B}\invmx{A}) =$ $\mx{A}(\mx{B}\invmx{B})\invmx{A} =$ $\mx{A}\mx{I}\invmx{A} =$ $\mx{A}\invmx{A} = \mx{I}$. Thus our guess was correct and the inverse of $(\mx{A}\mx{B})$ is indeed $\invmx{B}\invmx{A}$.

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As can be seen, we have $\mx{A}\mx{B}=\mx{O}$ and still, neither of $\mx{A}$ and $\mx{B}$ are $\mx{0}$. Note that this has some consequences that are not always intuitive.

It should be pointed out that sometimes it is more natural to use just as a row instead ofa transposed column. This is really up to the reader.

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It is also worth noting that Theorem 6/1 does notcontain the rule $\mx{A}\mx{B}=\mx{B}\mx{A}$ and the reason is that this is not truein most cases. This is shown in the following example.

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A square matrix may also be symmetric if it can be reflected along the main diagonal (from upper left down to lower right)while remaining the same. This is summarized in the following definition.

This can be solved to recover $x_1$, $x_2$, $x_3$ and $x_4$. Solving such is the topic of Chapter 5, which is about. Solving such was also the topic of chapter 8 of the Chinese textbook.