Math 2331 Linear Algebra

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1 1.7 Linear Independence Math 21 Linear Algebra 1.7 Linear Independence Shang-Huan Chiu Department of Mathematics, University of Houston math.uh.edu/ schiu/ February 5, 218 Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

2 1.7 Linear Independence Linear Independence and Homogeneous System Linear Independence: Definition Linear Independence of Matrix Columns Special Cases A Set of One Vector A Set of Two Vectors A Set Containing the Vector A Set Containing Too Many Vectors Characterization of Linearly Dependent Sets Theorem: Linear Dependence and Linear Combination Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

3 Linear Independence and Homogeneous System Example A homogeneous system such as x 1 x 2 x can be viewed as a vector equation 1 2 x 1 + x x 5 = = The vector equation has the trivial solution (x 1 =, x 2 =, x = ), but is this the only solution?. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, 218 / 17

4 Linear Independence: Definition Linear Independence A set of vectors {v 1, v 2,..., v p } in R n is said to be linearly independent if the vector equation has only the trivial solution. Linear Dpendence x 1 v 1 + x 2 v x p v p = The set {v 1, v 2,..., v p } is said to be linearly dependent if there exists weights c 1,..., c p,not all, such that c 1 v 1 + c 2 v c p v p =. linear dependence relation (when weights are not all zero) Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

5 Linear Independence: Example Example Let v 1 = 1 5, v 2 = 2 5, v = a. Determine if {v 1, v 2, v } is linearly independent. b. If possible, find a linear dependence relation among v 1, v 2, v. Solution: (a) x x x. = Augmented matrix: x is a free variable there are nontrivial solutions. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17.

6 Linear Independence: Example (cont.) {v 1, v 2, v } is a linearly dependent set (b) Reduced echelon form: 1 x 1 = 1 18 = x 2 = x Let x = (any nonzero number). Then x 1 = and x 2 = = or v 1 + v 2 + v = (one possible linear dependence relation) Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

7 Linear Independence of Matrix Columns Example (Linear Dependence Relation) can be written as the matrix equation: = 5 1 = Each linear dependence relation among the columns of A corresponds to a nontrivial solution to Ax =. The columns of matrix A are linearly independent if and only if the equation Ax = has only the trivial solution. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17.

8 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: {v 1 } The only solution to x 1 v 1 = is x 1 =. So {v 1 } is linearly independent when v 1. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

9 Special Cases: 2. A Set of Two Vectors Example (2. Let u 1 = A Set of Two Vectors) [ 2 1 ] [ 4, u 2 = 2 ] [ 2, v 1 = 1 ] [ 2, v 2 = a. Determine if {u 1, u 2 } is a linearly dependent set or a linearly independent set. b. Determine if {v 1, v 2 } is a linearly dependent set or a linearly independent set. Solution: (a) Notice that u 2 = u 1. Therefore ]. u 1 + u 2 = This means that {u 1, u 2 } is a linearly set. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, 218 / 17

10 Special Cases: 2. A Set of Two Vectors (cont.) (b) Suppose cv 1 + dv 2 =. Then v 1 = v 2 if c. But this is impossible since v 1 is a multiple of v 2 which means c =. Similarly, v 2 = v 1 if d. But this is impossible since v 2 is not a multiple of v 1 and so d =. This means that {v 1, v 2 } is a linearly set. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

11 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 Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

12 Special Cases:. A Set Containing the Vector Theorem A set of vectors S = {v 1, v 2,..., v p } in R n containing the zero vector is linearly dependent. Proof: Renumber the vectors so that v 1 =. Then v 1 + v v p = which shows that S is linearly. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

13 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 {v 1, v 2,..., v p } in R n is linearly dependent if p > n. Outline of Proof: Suppose p > n. A = [ v 1 v 2 v p ] is n p = Ax = has more variables than equations = Ax = has nontrivial solutions = columns of A are linearly dependent Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

14 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. a. 2 1, 6 4 b. Columns of Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

15 Special Cases: Examples (cont.) Examples (cont.) c. 2, 6, 1 d Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

16 Characterization of Linearly Dependent Sets Example Consider the set of vectors {v 1, v 2, v, v 4 } in R in the following diagram. Is the set linearly dependent? Explain Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17

17 Characterization of Linearly Dependent Sets Theorem An indexed set S = {v 1, v 2,..., v p } 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 v 1, then some vector v j (j 2) is a linear combination of the preceding vectors v 1,..., v j 1. Shang-Huan Chiu, University of Houston Math 21, Linear Algebra February 5, / 17