LA 1.3, 2.0 - 2.3, 3.0 - 3.3 Linear combinations, linear independence, span

Main points:

• Linear combination=a sum of multiples
• Linear equations: two interpretations
○ Following kind of a question: what multiplies of u and v can you add to get to w
○ Solving
§ Multiples are denoted as x and y
§ Solution is a pair (x,y) which satisfies the simultaneous system of two equations and two unknowns
• The geometry of linear equations
○ Points interpretation
○ Vector interpretation
• System of equations can have
○ No solution (two lines are parallel and not the same; and the two vectors are parallel and the vector we are trying to get to is not on that line)
○ One solution (two lines intersect in a point; and the two vectors can be combined in exactly one way to get to that vector)
○ Infinitely many solutions (two lines are the same and any point on this line works. The corresponding vectors are parallel and so is the vector we are trying to get to)
• Higher dimensions
• Matrix notation
○ Ax=b
○ An m * n matrix is a rectangular array of numbers, arranged in m rows and n columns
• Spanning
○ The span of a list of vectors is the set of all linear combinations that can be made
with those vectors
• Linear independence
○ No vector on the list is a linear combination of the other vectors on the list.
• Linear dependency
○ At least one of the vectors on the list is a linear combination of the other vectors on the list
• Subspaces and dimension
○ Subspace is a set of vectors that is the span of some list of vector
○ Dimension of a subspace is the minimum number of vectors required to span the subspace

Challenges:

I didn't understand what linear combinations are and what is their importance. I didn't understand what are the benefits and disadvantages of using vectors in contrast to system of equations. Not understanding linear combinations severely limited my understanding of spanning and rest of the topics. I am hoping to understand the things more clearly after tomorrow's class.

Reflection:

I was reminded how in math everything's builds on previously learnt things. As I didn't understand linear combinations, I had problems understanding spanning and the rest of the topics. I also came to realize how often I have had problems understanding the topics when I read them from the book, but how much easier it is for me to learn them in the classroom. For tomorrow, I'm especially interested to see how higher dimensions can be tried to "graph".