solution-sets-of-linear-systems.md (2049B)
1 +++ 2 title = 'Solution sets of linear systems' 3 template = 'page-math.html' 4 +++ 5 6 # Solution sets of linear systems 7 ## Homogeneous linear systems 8 homogeneous: if you can write it in $Ax = 0$ where A is an $m \times n$ matrix and 0 is the zero vector in $\Re^m$ 9 * always has at least one solution (the trivial solution, $x = 0$). 10 * has a nontrivial solution iff there is a free variable 11 * if the equation has only one free variable, the solution is a line through the origin 12 * when there are two or more free variables, it's a line through the origin 13 * solution set is $\text{Span} \{v_1, \ldots, v_p\}$ for suitable vectors 14 15 ## Parametric vector form 16 implicit description: 17 * a simple equation 18 * e.g. $10x_1 - 3x_2 - 2x_3 = 0$ 19 20 explicit description (parametric vector form): 21 * the solution to the equation as a set spanned by u and v 22 * of the form $x = su + tv$, with $s,t \in \Re$ 23 24 the solution set of $Ax = 0$ is $x = tv$ with $t \in \Re$. 25 26 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 $x = p + tv$ 27 28 Writing a solution set in parametric vector form: 29 1. Row reduce augmented matrix to echelon form 30 2. Express each basic variable in terms of any free variables. 31 3. Write a typical solution x as a vector, with entries depending on the (potential) free variables. 32 4. Decompose x into a linear combination of vectors using free vars as parameters. 33 34 ## Linear independence 35 36 linearly independent: 37 * set of vector equations: iff the vector equation has only the trivial solution ($x_1 = x_2 = x_3 = 0$) 38 * columns of matrix: iff $Ax = 0$ has _only_ the trivial solution 39 * one vector: iff v is not the zero vector 40 * two vectors: if neither of the vectors is a multiple of the other 41 42 linearly dependent: 43 * iff at least one of the vectors is a linear combination of the others 44 * if there are more vectors than entries in each vector 45 * if the set contains the zero vector 46 47 a set is linearly dependent iff it's not linearly independent. 48