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# Linear transformations as matrix vector products

Showing how ANY linear transformation can be represented as a matrix vector product. Created by Sal Khan.

## Want to join the conversation?

• I believe I have watched all videos up to this point and this is the first one that has confused me. It seems to jump straight into transforming a matrix whilst the overall subject has been transforming vectors. Whilst I understand that the matrix can be considered as a collection of column (or row) vectors it doesn't explain the apparent jump to matrix transformation in the usual thorough way. So whilst we started of transforming a vector we appear to have transformed a collection of vectors and used the result to transform the vector! What does the transformed matrix of basis vectors (the transformed I matrix) represent? • If any matrix-vector multiplication is a linear transformation then how can I interpret the general linear regression equation? `y = X β`.
X is the design matrix, β is a vector of the model's coefficients (one for each variable), and y is the vector of predicted outputs for each object.
Let's say X is a 100x2 matrix and β is a 2x1. Then y is a 100x1 matrix.

The concept is clear but from a Linear Transformation point of view what doest it mean?

I take a vector of coefficients in Rˆ2 and through X I transform it into a
vector in Rˆ100. I can't visualize it logically...🤔 • I'm confused as to what is being taught here. Is the the lesson saying that the transformation of a vector is equivalent to transforming the original basis and then using the result to transform the vector? • so, can i just arrange the linear 'instructions' in ascending order of the components of vector x take their coefficients of each term and plug it in to the matrix thats to be multiplied by the x vector ?? seems like a pretty legit shortcut now that i have an intuitive understanding of it • what do you call that matrix? Is it the standard matrix? • In the statement, from previous discussion I think "The sum equal to the sum of their transformation: " x1T(e1) + x2T(e2) + ...+ xnT(en)
should be written this way: e1T(x1) + e2T(x2) +...+ enT(xn). Can you explain why you put it that way and not like the way I thought it ought be written? Thanks • This series have been helping me a lot and I am thankful for it, but this one made me so confused to the point I got a little desperate. • At why does Sal want to avoid using L(x)? • At and forward, what was the purpose of the last example? I don't think I understand what it's supposed to prove.  