Main Points: Linear combinations involve adding multiples of two vectors, or summing multiples. Problems involving this idea includes ones that ask you to find what multiples of vectors u and v can make vector w. Using simultaneous equations, a solution can be found and plotted, where the the x and y values are the intersection of the two lines on the graph. Two fundamental but different geometric interpretations can be seen when looking at the equations ax + by = e and cx + dy = f. They are: 1. what point is on the intersection of lines? and 2. What multiples are added to give vector w? This entire idea of combinations works for dimensions higher than 2 as well. Matrices are lists of vectors, and one list of vectors is in one column. An m x n vector is a rectangular array of numbers arranged in m rows and n columns. Three main iddeas accompany vectors. First, the span of a set of vectors is the set of all linear combinations that can be made with those vectors. Second, if a vector on the listt is not a linear combination of the other vectors on that list, the vector is said to have linear independence. Finally, subspace is the set of vectors that is a span of some list of vectors. The dimension of subspace is the minimum number of vectors required to span the subspace. Subspace is a very special collection of vectors.
Challenges: I'm a bit confused about the matrices. Specifically, if you have 2 sets of vectors, such as
(a b) and (e f)
(c d) (g h)
can you multiply a and g and c and f? Also, can you still multiply c and d and g and h?
Reflections: I know I've stated this in previous blog entries, but I'm not quite sure how any of this actually fits into the real world. I don't really understand exactly what a matrix is either and how it is helpful in anything that we are doing.
Subscribe to:
Post Comments (Atom)
No comments:
Post a Comment