Linear Algebra Linear Independence

1.7 Linear Independence Math 2331 – Linear Algebra 1.7 Linear Independence Math 2331, Linear Algebra 1 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases 1.7 Linear Independence Linear Independence and Homogeneous System Linear Independence: Definition Linear Independence of Matrix Columns Special Cases A A A A Set Set Set Set of One Vector of Two Vectors Containing the 0 Vector Containing Too Many Vectors Characterization of Linearly Dependent Sets Theorem: Linear Dependence and Linear Combination Math 2331, Linear Algebra 2 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Linear Independence and Homogeneous System Example A homogeneous system such as      1 2 −3 x1 0  3 5     9 x2 = 0  5 9 3 x3 0 can be viewed as a vector equation         1 2 −3 0        9 = 0 . x1 3 + x2 5 + x3 5 9 3 0 The vector equation has the trivial solution (x1 = 0, x2 = 0, x3 = 0), but is this the only solution? Math 2331, Linear Algebra 3 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Linear Independence: Definition Linear Independence A set of vectors {v1 , v2 , . . . , vp } in Rn is said to be linearly independent if the vector equation x1 v1 + x2 v2 + · · · + xp vp = 0 has only the trivial solution. Linear Dpendence The set {v1 , v2 , . . . , vp } is said to be linearly dependent if there exists weights c1 , . . . , cp ,not all 0, such that c1 v1 + c2 v2 + · · · + cp vp = 0. ↑ linear dependence relation (when weights are not all zero) Math 2331, Linear Algebra 4 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Linear Independence: Example Example       1 2 −3 Let v1 =  3 , v2 =  5 , v3 =  9 . 5 9 3 a. Determine if {v1 , v2 , v3 } is linearly independent. b. If possible, find a linear dependence relation among v1 , v2 , v3 . Solution: (a)         1 2 −3 0        9 = 0 . x1 3 + x2 5 + x3 5 9 3 0 Augmented matrix:       1 2 −3 0 1 2 −3 0 1 2 −3 0  3 5 9 0  ∼  0 −1 18 0  ∼  0 −1 18 0  5 9 3 0 0 −1 18 0 0 0 0 0 x3 is a free variable ⇒ there are nontrivial solutions. Math 2331, Linear Algebra 5 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Linear Independence: Example (cont.) ⇒{v1 , v2 , v3 } is a linearly dependent set (b)  Reduced echelon  form: 1 0 33 0 x1  0 1 −18 0  =⇒ x2 0 0 0 0 x3 Let x3 = (any nonzero Then x1 = and x2 =   1  3 + 5 = = number). .   2  5 + 9     −3 0  9 = 0  3 0 or v1 + v2 + v3 = 0 (one possible linear dependence relation) Math 2331, Linear Algebra 6 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Linear Independence of Matrix Columns Example (Linear Dependence    1 −33  3  + 18  5 Relation)      2 −3 0 5  + 1 9  =  0  9 3 0 can be written as the matrix equation:      1 2 −3 −33 0  3 5 9   18  =  0 . 5 9 3 1 0 Each linear dependence relation among the columns of A corresponds to a nontrivial solution to Ax = 0. The columns of matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution. Math 2331, Linear Algebra 7 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: 1. A Set of One Vector Sometimes we can determine linear independence of a set with minimal effort. Example (1. A Set of One Vector) Consider the set containing one nonzero vector: {v1 } The only solution to x1 v1 = 0 is x1 = . So {v1 } is linearly independent when v1 6= 0. Math 2331, Linear Algebra 8 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: 2. A Set of Two Vectors Example (2. A Set of Two Vectors) Let  u1 =        2 4 2 2 , u2 = , v1 = , v2 = . 1 2 1 3 a. Determine if {u1 , u2 } is a linearly dependent set or a linearly independent set. b. Determine if {v1 , v2 } is a linearly dependent set or a linearly independent set. Solution: (a) Notice that u2 = u1 + u1 . Therefore u2 = 0 This means that {u1 , u2 } is a linearly Math 2331, Linear Algebra set. 9 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: 2. A Set of Two Vectors (cont.) (b) Suppose cv1 + dv2 = 0. Then v1 = v2 if c 6= 0. But this is impossible since v1 is a multiple of v2 which means c = . Similarly, v2 = v1 if d 6= 0. But this is impossible since v2 is not a multiple of v1 and so d = 0. This means that {v1 , v2 } is a linearly set. Math 2331, Linear Algebra 10 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: 2. A Set of Two Vectors (cont.) A set of two vectors is linearly dependent if at least one vector is a multiple of the other. A set of two vectors is linearly independent if and only if neither of the vectors is a multiple of the other. linearly linearly Math 2331, Linear Algebra 11 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: 3. A Set Containing the 0 Vector Theorem A set of vectors S = {v1 , v2 , . . . , vp } in Rn containing the zero vector is linearly dependent. Proof: Renumber the vectors so that v1 = v1 + v2 + · · · + which shows that S is linearly . Then vp = 0 . Math 2331, Linear Algebra 12 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: 4. A Set Containing Too Many Vectors Theorem If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. I.e. any set {v1 , v2 , . . . , vp } in Rn is linearly dependent if p > n. Outline of Proof: A=  v1 v2 · · · vp  is n × p Suppose p > n. =⇒ Ax = 0 has more variables than equations =⇒ Ax = 0 has nontrivial solutions =⇒columns of A are linearly dependent Math 2331, Linear Algebra 13 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: Examples Examples With the least amount of work possible, decide which of the following sets of vectors are linearly independent and give a reason for each answer.     9   3  2 , 6  a.   1 4  1  6 b. Columns of   9 4 2 7 8 3 3 8 7 2 4 9 6 1  5 0   5  8 Math 2331, Linear Algebra 14 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Special Cases: Examples (cont.) Examples (cont.)       9 0   3      2 , 6 , 0  c.   1 3 0   8       2  d.   1       4 Math 2331, Linear Algebra 15 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Characterization of Linearly Dependent Sets Example Consider the set of vectors {v1 , v2 , v3 , v4 } in R3 in the following diagram. Is the set linearly dependent? Explain Math 2331, Linear Algebra 16 / 17 1.7 Linear Independence Definition Matrix Columns Special Cases Characterization of Linearly Dependent Sets Theorem An indexed set S = {v1 , v2 , . . . , vp } of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent, and v1 6= 0, then some vector vj (j ≥ 2) is a linear combination of the preceding vectors v1 , . . . , vj−1 . Math 2331, Linear Algebra 17 / 17