We begin with simple properties of vectors: addition and scalar multiplication. I assume that you already know how to add two vectors e.g., ${\bf u} + {\bf v}$ and how to multiply a scalar to a vector e.g., $2 {\bf u}$. A little bit of thinking may give you the answer to the following questions: What does $c {\bf u} + d {\bf v}$ represent? or What does $c {\bf u} + d {\bf v} + e {\bf w}$ represent?, where $c, d, e$ are any numbers. These questions may look trivial but introduce you to some deeper concepts in linear systems.
Next, we will look into a central problem of linear algebra: when and how can we solve a linear system $A{\bf x} = {\bf b}$. where $A$ is a matrix and ${\bf x}$ and ${\bf b}$ are vectors. To answer to this question, we need to get used to some basic concepts in linear algebra. The first thing to learn is the geometric view of linear systems: we compare row pictures and column pictures of $A {\bf x}$. Using the column picture we can state whether the given system $A {\bf x} = {\bf b}$ is solvable.
Homework: Reading 1
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