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DEFINITION: A system of linear equations is said to be homogeneous if it can be written in the form Ax̄ = 0̄. Otherwise, it is nonhomogeneous. EXAMPLE: 3x1 + 5x2 = 0 6x1 + 2x2 = 0 HOMOGENEOUS 3x1 + 5x2 = 1 6x1 + 2x2 = 0 NONHOMOGEN. THEOREM: Suppose the equation Ax̄ = b̄ is consistent for some given b̄, and let p̄ be a solution. Then the solution set of Ax̄ = b̄ is the set of all vectors of the form w̄ = p̄ + v̄h, where v̄h is any solution of the homogeneous equation Ax̄ = 0̄. THEOREM: Let A be an m × n matrix. Then the following statements are logically equivalent: 1. A homogeneous system Ax̄ = 0̄ has only the trivial solution. 2. There are no free variables. 3. Number of columns of A = Number of pivot positions. DEFINITION: Vectors v̄1, . . . , v̄p are said to be linearly dependent if there exist scalars c1, . . . , cp, not all zero, such that c1v̄1 + . . . + cpv̄p = 0̄. Vectors v̄1, . . . , v̄p are said to be linearly independent if the vector equation c1v̄1 + . . . + cpv̄p = 0̄ has only the trivial solution. EXAMPLE: Vectors −2 1 v̄1 = 1 , v̄2 = −1 , 2 −4 are linearly dependent. Vectors 1 −2 v̄1 = 1 , v̄2 = −1 , 2 −4 are linearly independent. 3 v̄3 = 5 6 3 v̄3 = 5 7 THEOREM: Let A be an m × n matrix. Then the following statements are logically equivalent: 1. A homogeneous system Ax̄ = 0̄ has only the trivial solution. 2. There are no free variables. 3. Number of columns of A = Number of pivot positions. 4. The columns of a matrix A are linearly independent. THEOREM: Let A be an m × n matrix. Then the following statements are logically equivalent: 1. A homogeneous system Ax̄ = 0̄ has a nontrivial solution. 2. There are free variables. 3. Number of columns of A > Number of pivot positions. 4. The columns of a matrix A are linearly dependent. 5. At least one column of A is a linear combination of other columns. 1. Let 2. Let 0 0 1 v̄1 = 0 , v̄2 = 1 , v̄3 = 0 . 1 0 0 (a) Are v̄1, v̄2, v̄3 linearly independent? (b) Does {v̄1, v̄2, v̄3} span R3 ? 0 0 1 v̄1 = 0 , v̄2 = 1 , v̄3 = 2 . 0 0 0 (a) Are v̄1, v̄2, v̄3 linearly independent? (b) Does {v̄1, v̄2, v̄3} span R3 ? 3. Let 0 0 0 1 v̄1 = 0 , v̄2 = 1 , v̄3 = 0 , v̄4 = 0 . 2 1 0 0 (a) Are v̄1, v̄2, v̄3, v̄4 linearly independent? (b) Does {v̄1, v̄2, v̄3, v̄4} span R3 ? 4. Let 1 0 v̄1 = 0 , v̄2 = 0 . 0 1 (a) Are v̄1, v̄2 linearly independent? (b) Does {v̄1, v̄2} span R3 ? 1. Let 2. Let 9 7 5 v̄1 = 0 , v̄2 = 2 , v̄3 = 4 . −8 −6 0 (a) Are v̄1, v̄2, v̄3 linearly independent? (b) Does {v̄1, v̄2, v̄3} span R3 ? 9 7 5 4. v̄1 = 0 , v̄2 = 2 , v̄3 = −12 −6 0 (a) Are v̄1, v̄2, v̄3 linearly independent? (b) Does {v̄1, v̄2, v̄3} span R3 ? 3. Let 5 9 9 7 4. v̄1 = 0 , v̄2 = 2 , v̄3 = 4 , v̄4 = −12 −8 −6 0 (a) Are v̄1, v̄2, v̄3, v̄4 linearly independent? (b) Does {v̄1, v̄2, v̄3, v̄4} span R3 ? 4. Let 7 5 v̄1 = 0 , v̄2 = 2 . −6 0 (a) Are v̄1, v̄2 linearly independent? (b) Does {v̄1, v̄2} span R3 ?