What is the same in a diagram and a column calculation?
Switch between geometric motion and coordinate arithmetic to see one linear operation from two angles. Strang’s column combination is a worked example; neither representation is best for every task.
Read on through a shared source
What do the numbers in a vector describe? →Translate between an arrow and a list of coordinates. Each representation makes a different aspect of the same object easier to inspect.What does multiplying a vector mean? →Scaling one vector and combining two vectors are different operations. Check the operands and the type of result before interpreting the notation.What is geometric about a word embedding? →Coordinates describe a vector relative to a basis. Embeddings use learned vectors whose directions can capture relationships among words.
Compare the passages
the ability to translate back and forth between themthe geometric understanding is what lets you judge what tools to useA times x is a combination
of the columns of A.