Linear Algebra - 3

Section 1.5 Solution Sets of Linear Systems

Homogeneous Linear Systems :

• Homogeneous if it can be written in $Ax=0$
• Such $Ax=0$ always has at least one solution, called trivial solution
• $Ax=0$ has a nontrivial solution iff the equation has at least one free variable

Parametric Vector Form : $\vec x=s\vec u+t\vec v$

Solutions of Nonhomogeneous System : $\vec x=\vec p+t\vec v$. Note that $\vec p$ itself is just one particular solution.

Theorem : Suppose the equation $Ax=b$ is consistent for some given $b$, and let $p$ be a solution. Then the solution set of $Ax=b$ is the set of all vectors of the form $w=p+v_h$, where $v_h$ is any solution of the homogeneous equation $Ax=0$.

Section 1.7 Linear Independence

Definition of linear independence : the vector equation $x_1v_1+x_2v_2+\cdots+x_pv_p=0$ has only the trivial solution.

Linear dependence relation : $c_1v_1+c_2v_2+\cdots+c_nv_n=0$, where the weights $c_i$ are not all zero.

Linear Independence of Matrix Columns : iff $Ax=0$ only have trivial solution.

Sets of One, Two or More Vectors:

• One vector : is linearly independent iff $v$ is not the zero vector.
• Two vector : Just by inspection
• More vectors: set $S=\{v_1,v_2,\dots,v_n\}$ is linearly dependent iff at least one vectors in $S$ is a linear combination of the others. (Not for every vector)

Theorem : If a set contains more vectors than the entries in each vector, then the set is linearly dependent.

Theorem : If a set contains zero vector, then the set is linearly dependent.