Population Movement In 1990 the U.S. population, broken down...

Population Movement In 1990 the U.S. population, broken down by regions, was 50.8 million in the Northeast, 59.7 million in the Midwest. 85.4 million in the South. and 52.8 million in the West. $^{4}$ Between 1990 and 2000 the population in the Northeast grew by 2.8 million, the population in the Midwest grew by 4.7 million, the population in the South grew by 14.8 million, and the population in the West grew by 10.4 million. Set up the population figures for 1990 and the growth figures for the decade as row vectors. Assuming the same population growth from 2000 to 2010 as from 1990 to 2000 , use matrix operations to estimate the population in each region in $2010 .$ Compare the predicted 2010 population in the Northeast with the actual population given in Exercise $43 .$

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Key Concepts

Row Vectors

Row vectors are one-dimensional arrays that represent data in a single row. They are used to organize information such as population figures and growth values in a concise, ordered format, making it easier to perform linear algebra operations on the data.

Matrix Operations

Matrix operations, including addition and multiplication, allow for the manipulation of vectors and matrices to generate new data. In this context, they are used to add growth figures to the initial population vector and to perform transformations that model how the population changes over time.

Population Projections

Population projections involve using historical or current data to estimate future figures. By applying linear algebra techniques such as matrix operations to past growth data, one can create models that predict future population based on assumed growth rates.

Linear Growth Models

Linear growth models assume that the change in a quantity, like population, occurs at a constant rate over time. By using a fixed growth vector and adding it to the initial population vector, the model provides a simple yet powerful method for forecasting future population sizes.

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