For all Transf ∈ ℝn×n, the matrix is invertible if and only if rank(Transf) = n

The #determinant exists if and only if the transformation matrix is square.
The determinant in a linear transformation is the (signed) area of the image of the fundamental basis formed by the unit square.

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#Algebra thread 🧵

Which #matrices have an inverse?
Singular matrices never have an inverse.
When we look at the determinant, the determinant is non-zero for invertible matrices in the same way that non-zero numbers have an inverse.
Non-zero determinants mean that the matrices has an inverse, and a zero determinant means that the system (of sentences, of graphs) is singular.

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Data returned by an observation typically is represented as a vector in machine learning.

A neural network can be seen as a large collection of linear models. We may represent the inputs and outputs of each layer as vectors, matrices, and tensors (which are like higher dimensional matrices).

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Linear algebra (continued)

Which of the below operations, when applied to the rows of a matrix, keeps the #singularity (or non-singularity) of the matrix?:
(Hint: It works the same as a system of linear equations.)

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