Matrices are often presented as a useful bookkeeping/ commutation tools to students- but there’s much more to them. When you understand what a Matrix really is so many parts of Linear Algebra will be completely obvious to you… including the formula for matrix multiplication and the fact that matrices don’t commute. So here’s the big secret: A matrix is a linear transformation that eats a vectors and outputs another vector.

Homework questions:
Not all sized matrices can be multiplied together. Think about it in terms of them representing transformations from one space to another, and figure out which size matrices can be multiplied and explain why in the comments.
Consider a transformation that takes a 3d vector, and adds some fixed vector k to it. Say k is the vector 7 3 3. Is this a linear transformation or not? https://brilliant.org/practice/linear-transformations/?p=1
Imagine you have a matrix A that multiplies the first basis vector by 2, and the second basis vector by 6. How do you write A in this basis? https://brilliant.org/practice/linear-transformations/?p=3

Music: Epidemic sound, Summer nights 2

This video is an Introduction to Matrices but could be useful revision for school/university. If you have an exam, good luck!

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