[Sherman-Morrison formula, 4+4pt] Sometimes, one has to solve linear systems that are rank-1 modifications of other linear systems, for which a factorization is already available. Let us derive a solution algorithm for the modified system. Let A ∈ R n×n be invertible and u, v ∈ R n be column vectors. A + uvT is invertible with inverse (A + uvT ) −1 = A −1 − 1 1 + v TA−1u A −1uvTA −1 . if and only if v TA−1u 6= −1. (a) Let v TA−1u 6= −1 and assume given the LU decomposition of A. Specify an efficient algorithms based on the Sherman-Morrison formula to solve the rank-1 modified system (A + uvT )x = b for a given right hand side b

8 apples are left.

step-by-step explanation:

10-2=8

10 apples

take 2

now u have 8

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.