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October 2

Announcements

• Watch 2nd and 3rd 3blue1brown videos

• Section 1.1, 1.2 due this Thursday

• Worksheet 1 due Friday

• Section 2.1, 2.2 due next Thursday

• Office hours in Padelford C-8D

  • Wednesday 4:30 - 5:30

  • Thursday 12:00 - 1:00

Continue from last class

Justify the elementary row operations in class.

Gaussian elimination

Definition: The pivot positions are positions that contain a leading term. The pivot columns are columns that contain a pivot position. A pivot is the value of a pivot position.

Algorithm: Gaussian elimination is performed as follows:

• find the pivot position in the first row

• use elementary row operators to eliminate all value under the pivot position

• continue

work out example in class

Reduced echelon form

Definition: A matrix is in reduced echelon form if

• it is in echelon form

• all pivot positions contain a 1

• the only nonzero term in a pivot colum is in the pivot position

Algorithm: Gauss-Jordan elimination is performed as follows:

• do Gaussian elimination

• divide each row by the value of its pivot

• eliminate all other values in pivot column.

work out example in class.

Homogenous linear systems

A linear system is homogenous if the numbers to the right of the equal sign are all zero. They always have the trivial solution

2.1 Vectors

A vector is a list of number with addition and scalar multiplication defined. Given vectors $u=(u_1, u_2, \ldots, u_n)\in \mathbb{R}^n$, $v=(v_1,v_2,\ldots,v_n)\in \mathbb{R}^n$ of equal dimension and a scalar $c\in \mathbb{R}$, we define

• addition: $u+v=(u_1+v_1, u_2+v_2, \ldots, u_n+v_n)$,

• scalar multiplication: $cu=(cu_1,cu_2,\ldots,cu_n)$.

go over the geometry in class. tail to tip, parallelogram

Let $a,b$ be scalars and $u,v,w\in \mathbb{R}^n$. Then

• $u+v=v+u$,

• $a(u+v)=au+av$,

• $(a+b)u=au+bu$,

• $(u+v)+w = u+(v+w)$,

• $a(bu)=(ab)u$,

• $u+(-u)=0$,

• $u+0=0+u=u$,

• $1u=u$.

Definition: The If $u_1,u_2,\ldots,u_m$ are vectors and $c_1,c_2,\ldots,c_m$ are scalars, then $$c_1u_1+c_2u_2+\ldots+c_mu_m$$ is a linear combination of $u_1,\ldots,u_m$. Note that the constants can be negative or zero.

give examples in class.