Vectors | Chapter 1, Essence of linear algebra
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Further reading 5
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.How can columns explain a matrix equation? →Translate coordinates into a geometric picture. The columns of a matrix then show which combinations of vectors can reach the desired output.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.
Caption notes
Vectors can be arrows, coordinate lists or abstract objects. Addition and scalar multiplication connect geometric movement to numerical operations.
Original passages · 5
Every pair of numbers gives you one and only one vectorvector addition in this list of numbers conception looks like matching up their termsmultiplying a given vector by a scalar means multiplying each one of those componentsthe ability to translate back and forth between themthere are three distinct but related ideas about vectors