solution-sets-of-linear-systems.wiki (2069B)
1 == Solution sets of linear systems == 2 === Homogeneous linear systems === 3 homogeneous: if you can write it in $Ax = 0$ where A is an $m \times n$ matrix and 0 is the zero vector in $\Re^m$ 4 * always has at least one solution (the trivial solution, $x = 0$). 5 * has a nontrivial solution iff there is a free variable 6 * if the equation has only one free variable, the solution is a line through the origin 7 * when there are two or more free variables, it's a line through the origin 8 * solution set is $\text{Span} \{v_1, \ldots, v_p\}$ for suitable vectors 9 10 === Parametric vector form === 11 implicit description: 12 * a simple equation 13 * e.g. $10x_1 - 3x_2 - 2x_3 = 0$ 14 15 explicit description (parametric vector form): 16 * the solution to the equation as a set spanned by u and v 17 * of the form $x = su + tv$, with $s,t \in \Re$ 18 19 the solution set of $Ax = 0$ is $x = tv$ with $t \in \Re$. 20 21 if $Ax = b$ has a solution, then you get the solution set by translating the solution set of $Ax = 0$ using any particular solution p of $Ax = b$. The set is then $x = p + tv$ 22 23 Writing a solution set in parametric vector form: 24 1. Row reduce augmented matrix to echelon form 25 2. Express each basic variable in terms of any free variables. 26 3. Write a typical solution x as a vector, with entries depending on the (potential) free variables. 27 4. Decompose x into a linear combination of vectors using free vars as parameters. 28 29 === Linear independence === 30 31 linearly independent: 32 * set of vector equations: iff the vector equation has only the trivial solution ($x_1 = x_2 = x_3 = 0$) 33 * columns of matrix: iff $Ax = 0$ has _only_ the trivial solution 34 * one vector: iff v is not the zero vector 35 * two vectors: if neither of the vectors is a multiple of the other 36 37 linearly dependent: 38 * iff at least one of the vectors is a linear combination of the others 39 * if there are more vectors than entries in each vector 40 * if the set contains the zero vector 41 42 a set is linearly dependent iff it's not linearly independent. 43