91Ó°ÊÓ

Vector spaces are fundamental structures in linear algebra that provide a framework for working with vectors. In essence, a vector space is a collection of vectors where certain operations, like addition and scalar multiplication, are defined and obey specific rules (called axioms). These operations must satisfy properties such as:

• Closure under addition and scalar multiplication
• Associativity and commutativity of vector addition
• Existence of a zero vector
• Existence of additive inverses
• Distributivity of scalar multiplication with respect to vector addition and scalar addition

Every vector space has a basis, which is a set of linearly independent vectors that can be combined in various ways to form every other vector in that space. Understanding vector spaces is crucial because they appear in numerous areas of math and science, often serving as the stage upon which other concepts play out.

A linear combination is a way to create new vectors by combining existing ones in a vector space. In technical terms, a vector \(\mathbf{v}\) is a linear combination of vectors \(\mathbf{a}\) and \(\mathbf{b}\) if it can be expressed in the form \(\mathbf{v} = \alpha \mathbf{a} + \beta \mathbf{b}\), where \(\alpha\) and \(\beta\) are scalars. This idea is central in linear algebra because it allows us to understand how vectors relate to one another within a space.Key aspects of linear combinations include:

• They can be used to determine whether a set of vectors spans a space, meaning they can be combined to form any vector in that space.
• Learning about linear combinations leads to understanding linear independence and dependence, which are pivotal concepts for vector spaces.
• They are fundamental to solving systems of linear equations, where achieving a solution can be seen as finding specific linear combination of base vectors.

In our exercise, we verify that \(\mathbf{c}\) can be expressed as a linear combination of \(\mathbf{a}\) and \(\mathbf{b}\).

A system of equations is a set of equations with multiple variables. The goal is to find values for these variables that satisfy all the equations simultaneously. In linear algebra, a common form is a system of linear equations, which can often be expressed in matrix form.For example, our exercise involves a system of two equations:

• \(c_1 = \alpha a_1 + \beta b_1\)
• \(c_2 = \alpha a_2 + \beta b_2\)

The task is to find scalars \(\alpha\) and \(\beta\) that satisfy these equations, effectively expressing one vector as a linear combination of others. Solving such systems can be done using methods like substitution, elimination, or even more powerful matrix operations like Gaussian elimination.Understanding systems of equations is critical for many areas, as it enables analysis and solution-finding for complex problems across disciplines.

In linear algebra, determinants are scalar values that can be computed for square matrices. A determinant offers insights into several properties of a matrix, such as whether it is invertible or singular. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\).Key uses of determinants include:

• Determining the invertibility of a matrix: A non-zero determinant means the matrix is invertible.
• Computing the area or volume in geometry; determinants generalize these notions to higher dimensions.
• Solving systems of linear equations, as certain methods like Cramer's rule rely on determinants.

In our given exercise, the determinant \(a_1 b_2 - a_2 b_1\) being non-zero confirmed that the matrix formed by our vectors \(\mathbf{a}\) and \(\mathbf{b}\) is invertible, ensuring a unique solution for the scalars \(\alpha\) and \(\beta\). This concept is foundational for understanding how different mathematical structures interrelate.