A vector $v$ in two-dimensional space has two components $v_1$ and $v_2$.
$v + w = ( v_1 + w_1, v_2 + w_2)$ and $cv = ( cv_1, cv_2)$ are found a component at a time.
A linear combination of three vectors $u$ and $v$ and $w$ is $cu+ dv + ew$.
Take all linear combinations of $u$, or $u$ and $v$, or $u$, $v$, $w$. In three dimensions, those combinations typically fill a line, then a plane, then the whole space $R^3$.
Lengths and Dot Products
The dot product $v • w$ multiplies each component $v_i$ by $w_i$ and adds all $v_i w_i$.
The length $||v||$ is the square root of $v · v$. Then $u = \frac{v}{||v||}$ is a unit vector : length 1.
The dot product is v · w = 0 when vectors v and w are perpendicular.
The cosine of $\theta$ ( the angle between any nonzero v and w) never exceeds 1: Cosine $cos\theta=\frac{v · w}{||v||||w||}$, Schwarz inequality $|v · w|\leqq||v||||w||$.
Matrices
Matrix times vector: $Ax$ = combination of the columns of $A$.
The solution to $Ax = b$ is $x = A - lb$, when $A$ is an invertible matrix.
The cyclic matrix $C$ has no inverse. Its three columns lie in the same plane. Those dependent columns add to the zero vector. $Cx = 0$ has many solutions.
This section is looking ahead to key ideas, not fully explained yet.
Chapter 2 Solving Linear Equations
Vectors and Linear Equations
The basic operations on vectors are multiplication cv and vector addition v + w.
Together those operations give linear combinations cv + dw.
Matrix-vector multiplication Ax can be computed by dot products, a row at a time. But Ax must be understood as a combination of the columns of A.
Column picture: Ax = b asks for a combination of columns to produce b.
Row picture: Each equation in Ax = b gives a line (n = 2) or a plane (n = 3) or a “hyperplane” (n > 3). They intersect at the solution or solutions, if any.