â SpanÖ?ß@ œ Ö =? > @ À =ß> real numbers : SpanÖ?ß@ œ Ö: =? > @ À =ß> real numbers œ the previous plane with each point translated by : Ðfor example, is translated to :) á In general: Adding a vector : to each vector in a set translates the set of vectors along a copy of the vector :
Theorem Suppose E, is consistent. Let : be one particular solution (so that E: œ, Ñ Then every solution ; for E, can be written as ; œ : 2 Å where 2 is a solution of the corresponding homogeneous equation E More informally: when E, is consistent, you can write all its solutions by taking a particular solution : of the equation and adding on all possible solutions of the corresponding homogenous equation E Þ If is happens that E has only one solution,, then E, also has only one solution, : œ : Þ (See proof in class notes or text.)
Exercises 1. E Þ Suppose the rref for the coefficient matrix E is Ô " # " " " Õ The general solution in parametric vector form is:??? Solution: The rref for the coefficient matrix for E is: rrefðeñ œ Ô " # " " " Õ #B B# œ B# B " # B B $ % % % B Parametric vector form Ô B" Ô # Ô Ô " B# " B$ œ B# B% " B Ö Ù Ö Ù Ö Ù Ö Ù B% " ÕB Õ Õ Õ "
2. E, Ðsame matrix E as in 1)). Suppose the rref for the augmented matrix is Ô " # " l $ " " l % Õ l 0 Write solution in parametric vector form: Solution:rref of augment matrix œ Ô " # " l $ " " l % Õ l 0 B" œ $ #B# B B# œ B# B$ œ % B% B% œ B% B, so (free) (free) (free) Ô $ Ô # Ô Ô " " % B# B% " B Ö Ù Ö Ù Ö Ù Ö Ù " Õ Õ Õ Õ " Å Å a particular general solution for solution of E, homogeneous system E Look at the relationship between the solution sets in 1. and 2. Ô $ Note that this set of solutions is a translation along a copy of the vector % of the Ö Ù Õ solution set in the preceding problem 1. 3. E,. What is the general solution if the rref for the Ô " # " $ " " %? Õ augmented matrix is
The set of vectors Ö@ ß@ ß ÞÞÞß @ is linearly dependent if " # : if B" @" B# @# ÞÞÞ B: @: œ has a nontrivial solution, that is there are scalars - ß- ßÞÞÞß- " # : not all - @ - @ ÞÞÞ - @ œ " " # # : : Same as saying homogeneous matrix equation for which E Ô B" B# Ò@ "@# ÞÞÞ@ : ÓÖ Ù œ has a nontrivial solution ã ÕB : (which happens if E has free variables) Otherwise, we say that the set Ö@ " ß@ # ßÞÞÞß @: is linearly independent Sometimes we say that the vectors @" ß@ # ßÞÞÞß @: are linearly dependent or independent. Ô $ Ô # Ô " Example Is ', ), # Ÿ Õ$ Õ Õ $ linearly independent? We know how to do the calculation to get the answer. You just have know linearly independent means Look at homogeneous vector equation Ô $ Ô # Ô " Ô B" ' B# ) B# # œ Õ$ Õ Õ $ Õ @" @# @ $ or
equivalent matrix equation Ô $ # " ' ) # Õ$ $ Does the equation have only the trivial solution (then, the set is linearly independent)? or are there aksi other solutions (then, the set is linearly dependent)? Since @" @# @ $ Ô $ # " Ô " " ' ) # µ ÞÞÞ µ " " Õ$ $ Õ there are infinitely many solutions: B" œ B$ B# œ B$ B$ œ B$ So can use weights (for example) "ß " $ " # The set is linearly dependent. Check: for example, Ô $ Ô # Ô " Ô " ' " ) Ð "Ñ # œ Õ$ Õ Õ $ Õ Ô 1 Ô 0 Ô 2 0 1 1 Example Is Ö Ù, Ö Ù, Ö ÙŸ linearly independent? 1 0 0 Õ 1 Õ 0 Õ 0 Homogeneous vector equation Ô 1 Ô 0 Ô 2 Ô 0 0 1 1 0 B" Ö Ù B# Ö Ù B$ Ö Ù œ Ö Ù 1 0 0 0 Õ 1 Õ 0 Õ 0 Õ 0 equivalent matrix equation Ô " # " " Ö Ù " Õ "
Row reduction of coefficient matrix: Ô " # Ô " " " " Ö Ù µ ÞÞÞ µ Ö Ù " " Õ" Õ Answer to original question:? Important Results about linear dependence and independence (for vectors in any 8 Ñ See proofs from class or textbook. 1. Ð For : 2) Ö@ " ßÞÞÞß @: is linearly dependent if and only if at least one of the vectors is a linear combination of the other vectors. 2. Ö is linearly dependent, but Ö @ is linearly independent if @ Á " "