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  "contentMarkdown": "# Linear Algebra - 基变换\n> 创建时间：2020/11/16 17:54\n\n## Linear Algebra - 基变换\n\nAnyway to **translate** between vectors and sets of numbers is called a **coordinate system**.\n\n![Alt text](assets/1605518712512.png)\n\n矩阵向量乘法（matrix-vector multiplication）左乘的本质：\n\n![Alt text](assets/1605519239822.png)\n\n得到的是，之前坐标系下坐标描述的向量 在我们的坐标系下的表示\n\n线性变换矩阵 - 列为坐标系A下的基向量\n\n  1. Geometrically：\nThis matrix transform A坐标系下的网格(grid) into B坐标系下的网格\n\n  2. Numerically：\ntranslating a vector described in B坐标系 to A坐标系下的坐标\n\n  3. 理解：\n把A坐标系下的拥有相同坐标的向量（A坐标系下对B坐标系下相同坐标向量的 **误解 misconception** ）变换为实际描述的向量。\n\n**逆矩阵** \\- The inverse of a transformation is a new transformation that corresponds to(相当于) playing that first one backwards.\n将所对应的变换逆向进行。\n\n坐标变换矩阵：\n一个矩阵的列代标B坐标系的基向量，却是用A坐标系的坐标来描述。\n\n矩阵M表示在A坐标系下的某个变换。对B坐标系下的坐标进行相同变换：\n\n![Alt text](assets/1605521091498.png)\n\n转换成A坐标系下的坐标，变换完后再转回去。\n\n以下公式即为将A坐标系下的线性变换矩阵 ，变为相同线性变换在B坐标系下的矩阵表示：\n\n![Alt text](assets/1605521149213.png)\n\n线性变换：\n\n  1. 向量的变换\n\n  2. 不同空间下的坐标转换\n\n  3. 变换矩阵的转换\n\n![Alt text](assets/1605521372447.png)\n\n视角上的转移——对同一个变换，在不同坐标空间下观察。\n"
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