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Math1300:MainPage/LinearIndependence Contents • 1 Linear Combinations and Linearly Independent Sets of Vectors ♦ 1.1 Definition of Linear Independence ♦ 1.2 How to prove that a set of vectors is linearly independent ♦ 1.3 Examples of Linear Independence (Polynomials) ♦ 1.4 Examples of Linear Independence (Euclidean n-space) ♦ 1.5 Examples of Linear Independence (Matrices) ♦ 1.6 Small Linearly Independent Sets ◊ 1.6.1 Theorem (Independent Sets with Two Vectors) ◊ 1.6.2 Theorem (Sets containing 0 are dependent) ♦ 1.7 Properties of Linearly Independent Sets ◊ 1.7.1 Theorem (one vector as a combination of others) ◊ 1.7.2 Corollary (one vector as a combination of others) ◊ 1.7.3 Theorem (Large sets in n-space are linearly dependent) ◊ 1.7.4 Theorem (Uniqueness of the Linear Combination) Linear Combinations and Linearly Independent Sets of Vectors Definition of Linear Independence Suppose is a set vectors in the vector space V, and consider the linear combination then The set S is linearly independent if this is the only way a linear combination can equal If A set of vectors in a vector space V is linearly independent if implies How to prove that a set of vectors is linearly independent If Contents is the set of vectors, set and prove that This will, by definition, make the set of vectors linearly independent. 1 Math1300:MainPage/LinearIndependence Examples of Linear Independence (Polynomials) • Let p(x) = x2 − 3x + 2 and q(x) = 2x2 − 1. To see if S = {p(x),q(x)} is linearly independent, we set a linear combination of the vectors equal to This gives us a set of equations, one for each power of x: and so r1 = r2 = 0. This means that S is linearly independent. • Let p1(x) = x2 − 3x + 2, p2(x) = 2x2 − 1 and p3(x) = x2 + 3x − 3. To see if S = {p1(x),p2(x),p3(x)} is linearly independent, we set a linear combination of the vectors equal to zero. This gives the system of linear equations with augmented matrix and reduced row echelon form which means the solutions are (r ,r ,r ) = (t, − t,t) with Hence there are values of r1,r2,r3, not all zero, so that This means that {p (x),p (x),p (x)} is not linearly 1 in turn 2 3 independent. Indeed, if t = 1, then r1 = 1, r2 = − 1 and r3 = 1, which gives: 1 2 3 Examples of Linear Independence (Euclidean n-space) • Let v = (1,2,3,1) and v = (3,1,0,2). To test for linear independence, we use 1 (0,0,0,0) = r (1,2,3,1) 2 + r2(3,1,0,2) = (r1 + 3r2,2r1 + r2,3r1,r1 + 2r2). 1 Each coordinate contributes an equation to the system: and so r1 = r2 = 0 and the set S = {v1,v2} is linearly independent. • Let S = {(1,2,3,1),(3,1,0,2),( − 1,3,6,0)}. To test for linear independence we use the equation Examples of Linear Independence (Polynomials) 2 Math1300:MainPage/LinearIndependence As before, each coordinate contributes an equation to the system: This system of equations has augmented matrix which has reduced row echelon form and hence we have (r1,r2,r3) = ( − 2t,t,t) for all Since the equation is valid when r1 = r2 = r3 = 0 does not hold, the set S is linearly dependent. Examples of Linear Independence (Matrices) • Let To see if S is linearly independent, we use the equation which corresponds to four equations These equations have only r1 = r2 = r3 = 0 as a solution. This means that S is linearly independent. • Let To see if S is linearly independent, we use the equation which corresponds to four equations Examples of Linear Independence (Euclidean n-space) 3 Math1300:MainPage/LinearIndependence which has augmented matrix which has reduced row echelon form and so (r1,r2,r3) = ( − t, − t,t) for In particular, there exist solutions to the system of linear equations where not all ri = 0. Hence S is linearly dependent. In particular, when t = 1 we have Small Linearly Independent Sets If and dependent. then, setting r = 1, we have and so S is linearly 1 If and independent. then, as we saw here If implies r = 0. This means that S is linearly is linearly dependent, then nonzero. Say that scalar multiple of dependent. Then that is, has a solution with at least one of r ,r 1 2 and so then is a scalar multiple of and so Conversely, if is a is linearly Theorem (Independent Sets with Two Vectors) is linearly dependent if and only if one vector is a scalar multiple of the other. Theorem (Sets containing 0 are dependent) If a set of vectors S contains Proof Let linearly dependent. then S is linearly dependent. Then, using r = 1, we have Examples of Linear Independence (Matrices) 1 and so S is 4 Math1300:MainPage/LinearIndependence Properties of Linearly Independent Sets Theorem (one vector as a combination of others) A set with k > 1 is linearly dependent if and only if some vector in S is a linear combination of the other vectors in S. Proof If S is linearly dependent, then Then with not all ri = 0. Say that and and is a linear combination of the other vectors of S. Conversely, suppose that dependent (at least one is a linear combination of the other vectors of S. This means which in turn implies which makes S linearly since r = − 1). j Corollary (one vector as a combination of others) A set with k > 1 is linearly independent if and only if no vector in S is a linear combination of the other vectors in S. Theorem (Large sets in n-space are linearly dependent) Let be a set of vectors in and assume that k > n. Then S is linearly dependent. Proof The equation gives a system of homogenious linear equations, one for each coordinate. Hence there are n equations with as k unknowns. as we have proven this system in fact has an infinite number of nontrivial solutions, and so S is linearly dependent. Theorem (Uniqueness of the Linear Combination) Let be a linearly independent set of vectors in V and can be written as a linear combination of in at most one way. Proof Suppose Since S is linearly independent, we have Properties of Linearly Independent Sets be a vector in V. Then and Then and so 5