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Somos
Somos
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Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.

For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for anthe equivalence class of the vector with coordinates $(x,y,z).$

Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.

For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for an equivalence class of the vector with coordinates $(x,y,z).$

Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.

For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for the equivalence class of the vector with coordinates $(x,y,z).$

Added explicit 3x3 example.
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Somos
Somos
  • 2.8k
  • 12
  • 20

Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.

For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for an equivalence class of the vector with coordinates $(x,y,z).$

Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.

Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.

For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for an equivalence class of the vector with coordinates $(x,y,z).$

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Somos
Somos
  • 2.8k
  • 12
  • 20

Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.