By Stanley I. Grossman ; traductor: Alfonso Leal Guajardo ; revisor tecnico: Francisco Paniagua Bocanegra.
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That is, 2 3 2 3 2 3 1 2 7 34 2 5 C 24 5 5 D 4 4 5 5 6 3 Observe in Example 5 that the original vectors a1 , a2 , and b are the columns of the augmented matrix that we row reduced: 2 3 1 2 7 4 2 5 45 5 6 3 ✻ a1 ✻ ✻ a2 b For brevity, write this matrix in a way that identiﬁes its columns—namely, Œ a1 a2 b (4) It is clear how to write this augmented matrix immediately from vector equation (1), without going through the intermediate steps of Example 5. Take the vectors in the order in which they appear in (1) and put them into the columns of a matrix as in (4).
The points in the plane corresponding to and 5 Ä 5 lie on a line through the origin. 2 c. An example of a linear combination of vectors v1 and v2 is the vector 12 v1 . d. The solution set of the linear system whose augmented matrix is Œ a1 a2 a3 b is the same as the solution set of the equation x1 a1 C x2 a2 C x3 a3 D b. e. The set Span fu; vg is always visualized as a plane through the origin. 24. a. Any list of ﬁve real numbers is a vector in R5 . b. The vector u results when a vector u vector v.
9/ row 2 Finally, scale row 1, dividing by the pivot, 3. 2 1 40 0 0 1 0 2 2 0 3 2 0 0 0 1 3 24 75 4 ✛ Row scaled by 1 3 This is the reduced echelon form of the original matrix. The combination of steps 1–4 is called the forward phase of the row reduction algorithm. Step 5, which produces the unique reduced echelon form, is called the backward phase. NUMERICAL NOTE In step 2 above, a computer program usually selects as a pivot the entry in a column having the largest absolute value. This strategy, called partial pivoting, is used because it reduces roundoff errors in the calculations.
Aplicaciones de algebra lineal by Stanley I. Grossman ; traductor: Alfonso Leal Guajardo ; revisor tecnico: Francisco Paniagua Bocanegra.