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This is just a collections of specific terms in linear algebra and matrix computations. Instead of providing the calculation function, it just provides a direct view to explain these terms.

• span: The “span” of $$\vec v$$ and $$\vec w$$ is the set of all the linear combinations.
$a\vec v + v\vec w$

where $$a$$ and $$b$$ vary over all real numbers.

• linear transformation: compositions of “rotation” and “shear” operation. Applying one transformation after another.

• Determinant: to measure the fact of by which the given region increase or decreases.
• $$det(M)=0$$ mean it squash the matrix into lower dimension.
• negative determinant: flip over the orientation of your axes.
• Duality:
• dot product $$\leftrightarrow$$ matrix-vector product.
$\begin{bmatrix} u_x &u_y\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=u_x x+u_y y$ $\begin{bmatrix} u_x \\ u_y\end{bmatrix}\cdot \begin{bmatrix}x\\ y\end{bmatrix}=u_x x+u_y y$
• cross product:

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