What exactly is a matrix?

I know how basic operations are performed on matrices, I can do transformations, find inverses, etc. But now that I think about it, I actually don’t “understand” or know what I’ve been doing all this time. Our teacher made us memorise some rules and I’ve been following it like a machine.

  1. So what exactly is a matrix? And what is a determinant?

  2. What do they represent?

  3. Is there a geometrical interpretation?

  4. How are they used? Or, rather, what for are they used?

  5. How do I understand the “properties” of matrix?

I just don’t wanna mindlessly cram all those properties, I want to understand them better.

Any links, which would improve my understanding towards determinants and matrices? Please use simpler words. Thanks 🙂

Answer

A matrix is a compact but general way to represent any linear transform. (Linearity means that the image of a sum is the sum of the images.) Examples of linear transforms are rotations, scalings, projections. They map points/lines/planes to point/lines /planes.

So a linear transform can be represented by an array of coefficients. The size of the matrix tells you the number of dimension of the domain and the image spaces. The composition of two linear transforms corresponds to the product of their matrices. The inverse of a linear transform corresponds to the matrix inverse.

A determinant measures the volume of the image of a unit cube by the transformation; it is a single number. (When the number of dimensions of the domain and image differ, this volume is zero, so that such “determinants” are never considered.) For instance, a rotation preserves the volumes, so that the determinant of a rotation matrix is always 1. When a determinant is zero, the linear transform is “singular”, which means that it loses some dimensions (the transformed volume is flat), and cannot be inverted.

The determinants are a fundamental tool in the resolution of systems of linear equations.

As you will later learn, a linear transformation can be decomposed in a pure rotation, a pure (anisotropic) scaling and another pure rotation. Only the scaling deforms the volumes, and the determinant of the transform is the product of the scaling coefficients.

Attribution
Source : Link , Question Author : William , Answer Author : Community

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