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A linear combination of vectors refers to an expression composed of sums and scalar multiples of these vectors. In simpler terms, you can imagine it as a recipe where you mix or combine different vectors using certain coefficients. These coefficients are generally numbers that determine how much each vector contributes to the final combination. For example, consider vectors \( \mathbf{u}_1 = (1, 0) \) and \( \mathbf{u}_2 = (0, 1) \). Any vector \( \mathbf{w} = (a, b) \) in this space can be expressed as a linear combination of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) by finding coefficients \( c_1 \) and \( c_2 \) such that \( \mathbf{w} = c_1 \cdot \mathbf{u}_1 + c_2 \cdot \mathbf{u}_2 \). Here, \( c_1 \) and \( c_2 \) are the coefficients, and their values ensure that \( \mathbf{w} \) is created as a combination of these two basis vectors.
This is a critical concept because it shows us how we can decompose a vector into simpler parts.

When expressing a vector as a linear combination of basis vectors, we often have to solve systems of equations to determine the necessary coefficients. Each equation in the system corresponds to one of the vector components and these equations must be solved in such a way that the combination results in the desired vector.For instance, if we have a vector \( \mathbf{w} = (1, 1) \) and we aim to express this as a combination of \( \mathbf{u}_1 = (2, -4) \) and \( \mathbf{u}_2 = (3, 8) \), we need to solve the system:

• \( 2c_1 + 3c_2 = 1 \)
• \( -4c_1 + 8c_2 = 1 \)

Solving these equations gives us the coefficients \( c_1 \) and \( c_2 \), which tell us exactly how much of each basis vector we need to mix together to form \( \mathbf{w} \).
This process is not only about finding the coefficients but also about understanding relationships among all possible vectors in the vector space.

Basis vectors are a set of vectors in a vector space that are linearly independent and span the vector space. This means they are like building blocks that you can use to express any vector in that space. The key characteristics of basis vectors are:

• They must be linearly independent – no vector in the set can be expressed as a linear combination of the others.
• They must span the vector space – any vector in the space can be expressed as a combination of the basis vectors.

For example, in the 2-dimensional real number space (\(R^2\)), the standard basis vectors are \( \mathbf{u}_1 = (1, 0) \) and \( \mathbf{u}_2 = (0, 1) \). Every vector in \(R^2\) can be expressed using these two vectors.This is critical because, with the right choice of basis vectors, finding the coordinate vector (or expressing vectors as a linear combination) becomes significantly simpler.

A vector space is a collection of vectors that can be added together and multiplied by scalars while still remaining within the same space. Vector spaces are fundamental in the field of linear algebra and can be visualized as the entire set of possible linear combinations of a given set of vectors.In the given exercise, we're working in the vector space \(R^2\), which is essentially a plane consisting of all 2-dimensional vectors. A vector space must satisfy certain properties including:

• Closure under vector addition
• Closure under scalar multiplication
• Contains a zero vector
• Each vector has an additive inverse

Understanding vector spaces is important because they provide the framework within which we perform operations like finding coordinate vectors, defining basis vectors, and expressing vectors as linear combinations.
These spaces help mathematicians and engineers solve real-world problems by mapping complex scenarios into simple, manageable mathematical models.