From the course: Machine Learning Foundations: Linear Algebra
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Basis, linear independence, and span
From the course: Machine Learning Foundations: Linear Algebra
Basis, linear independence, and span
- [Instructor] We can build up every vector in the vector space from the elements in a spanning set using only the operations of addition and scalar multiplication. By definition, a spanning set is: "The set v1 until vn is a spanning set for V "if, and only if, every vector in V "can be written as a linear combination "of v1, v2, until vn." Let's explore spanning set with a few simple examples. If we draw a single nonzero vector, v1, then the span consists of all vectors of the form lambda1 a1. Lambda1 can be positive, negative, or zero. Say if you take a multiple of v1, you can get anywhere along the one-dimensional space of a line. As you can see, for any point not on that line, the corresponding vector will not be in the span v1. If we want to span the entire space, we'll need at least two vectors. The easiest way to do that is by selecting the basis vector e1 that is equal 1,0 and e2 that is equal to 0,1.…
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