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Extra resources for A Guide to Arithmetic [Lecture notes]

Example text

Row reduce the system’s augmented matrix: 2 1 4 2 3 2 4 6 1 5 6 3 5 8 3 0 35 2 2 1 40 0 2 1 40 0 2 0 0 1 3 3 3 1 1 2 0 0 1 3 0 3 1 0 3 0 35 2 3 0 35 5 This echelon matrix shows that the system is inconsistent, because its rightmost column is a pivot column; the third row corresponds to the equation 0 = 5. There is no need to perform any more row operations. Note that the presence of the free variables in this problem is irrelevant because the system is inconsistent. 3. Since the coefficient matrix has four pivots, there is a pivot in every row of the coefficient matrix.

V1 C u1 ; : : : ; vn C un / DvCu hϭ 9 v2 v1 hϭ 5 Span {v1, v2, v3} The points Ϫ4 3 h v3 hϭ 1 lie on a line that intersects the plane when h ϭ 5. Definition of vector addition Commutativity of addition in R Definition of vector addition 2. The vector y belongs to Span fv1 ; v2 ; v3 g if and only if there exist scalars x1 ; x2 ; x3 such that 2 3 2 3 2 3 2 3 1 5 3 4 x1 4 1 5 C x2 4 4 5 C x3 4 1 5 D 4 3 5 2 7 0 h This vector equation is equivalent to a system of three linear equations in three unknowns.

Why (or why not)? True/false questions of this type will appear in many sections. 1. 2 25. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent. 26. Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution. 27. ” 28. What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

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