A linear combination is a way of combining multiple mathematical objects, such as vectors or functions, by multiplying each object by a scalar and then adding the results together. This concept is particularly important in the context of solving systems of equations by elimination.
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A linear combination of vectors $\vec{a}$ and $\vec{b}$ is given by the expression $c_1\vec{a} + c_2\vec{b}$, where $c_1$ and $c_2$ are scalars.
Linear combinations are used to represent a vector as a sum of other vectors, which is crucial for solving systems of equations by elimination.
The coefficients $c_1$ and $c_2$ in a linear combination determine the relative contribution of each vector to the final result.
Linear combinations can be used to eliminate variables in a system of equations by setting the coefficients of the variable to be eliminated equal to zero.
The ability to express one vector as a linear combination of others is a fundamental concept in linear algebra and has many applications in mathematics and science.
Review Questions
Explain how linear combinations are used to solve systems of equations by elimination.
To solve a system of equations by elimination, we can use linear combinations to eliminate one of the variables. This is done by multiplying one equation by a scalar and adding or subtracting it from another equation, causing the coefficient of the variable to be eliminated to become zero. The resulting equation can then be solved for the remaining variable, and the solution can be substituted back into the original equations to find the values of the other variables.
Describe how the coefficients in a linear combination affect the final result.
The coefficients $c_1$ and $c_2$ in a linear combination $c_1\vec{a} + c_2\vec{b}$ determine the relative contribution of each vector $\vec{a}$ and $\vec{b}$ to the final result. Changing the values of these coefficients will change the direction and magnitude of the resulting vector, allowing us to manipulate the linear combination to achieve a desired outcome, such as eliminating a variable in a system of equations.
Analyze the role of linear combinations in the broader context of linear algebra and its applications.
Linear combinations are a fundamental concept in linear algebra, as they allow us to represent vectors as a sum of other vectors. This is crucial for understanding the properties of vector spaces and the relationships between vectors. Linear combinations have numerous applications in mathematics and science, such as in the analysis of data, the solution of differential equations, and the study of quantum mechanics. The ability to express one vector as a linear combination of others is a powerful tool that underpins many important mathematical and scientific theories and techniques.