A linear combination is the sum of a set of vectors, each multiplied by a corresponding scalar (numerical) coefficient. It represents a way of combining multiple vectors into a single vector by applying specific weights or coefficients to each vector.
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In the context of vectors, a linear combination allows you to create new vectors by adding and scaling existing vectors.
When solving systems of linear equations with three variables, linear combinations of the equations can be used to eliminate variables and simplify the system.
The coefficients in a linear combination represent the weights or proportions of each vector or equation being combined.
Linear combinations are a fundamental concept in linear algebra and are used extensively in various mathematical and scientific applications.
The ability to form linear combinations is crucial for understanding the properties of vector spaces and for solving complex systems of equations.
Review Questions
Explain how linear combinations can be used to represent new vectors in the context of 10.8 Vectors.
In the context of 10.8 Vectors, linear combinations allow you to create new vectors by adding and scaling existing vectors. For example, if you have two vectors, $\vec{u}$ and $\vec{v}$, you can form a linear combination $a\vec{u} + b\vec{v}$, where $a$ and $b$ are scalar coefficients. This new vector is a weighted sum of the original vectors, and its magnitude and direction are determined by the values of $a$ and $b$. Linear combinations of vectors are a fundamental way to manipulate and combine vector quantities in geometric and algebraic applications.
Describe how linear combinations can be used to solve systems of linear equations with three variables, as discussed in 11.2 Systems of Linear Equations: Three Variables.
In the context of 11.2 Systems of Linear Equations: Three Variables, linear combinations can be used to simplify and solve systems of linear equations. When dealing with a system of three equations and three variables, you can apply linear combinations to the equations to eliminate one of the variables. For example, by taking a linear combination of two of the equations, you can cancel out one of the variables, reducing the system to two equations in two variables. This process can be repeated to eventually solve for the values of all three variables. The ability to form strategic linear combinations of the equations is a key technique for solving complex systems of linear equations in multiple variables.
Analyze how the properties of linear combinations, such as the ability to scale and add vectors, contribute to the broader understanding of vector spaces and their applications.
The properties of linear combinations, such as the ability to scale and add vectors, are fundamental to the concept of vector spaces, which is a central topic in linear algebra. Vector spaces are defined by their ability to support linear combinations of their elements, which in this case are vectors. The fact that linear combinations of vectors result in new vectors that are also part of the vector space demonstrates the closed nature of vector spaces under these operations. This allows for the development of a rich mathematical theory and the application of vector spaces in various fields, including physics, engineering, and computer science. Understanding linear combinations is crucial for grasping the underlying structure and properties of vector spaces, which have far-reaching implications in both theoretical and practical domains.
A system of linear equations is a set of two or more linear equations that share common variables and must be solved simultaneously to find the values of those variables.