26. Vector Spaces
Vector spaces serve as a core component of linear algebra, instrumental in various fields of Civil Engineering such as structural analysis and hydraulics. This chapter elucidates the definitions, properties, and applications of vector spaces, equipping students with essential mathematical reasoning for tackling complex engineering problems. Key concepts include linear combinations, independence, bases, transformations, and practical applications in engineering contexts.
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What we have learnt
- A vector space is defined by vector addition and scalar multiplication under specific axioms.
- Subspaces, linear combinations, and spans are fundamental concepts that extend the properties of vector spaces.
- Applications of vector spaces in Civil Engineering range from structural modeling to computational tools in various software programs.
Key Concepts
- -- Vector Space
- A set equipped with vector addition and scalar multiplication that satisfies certain axioms.
- -- Linear Independence
- A set of vectors is linearly independent if no vector can be expressed as a linear combination of the others.
- -- Basis
- A linearly independent set of vectors in a vector space that spans the entire space.
- -- Dimension
- The number of vectors in a basis of a vector space, indicating its size.
- -- Linear Transformation
- A mapping between vector spaces that preserves vector addition and scalar multiplication.
- -- Inner Product Space
- A vector space with an inner product that enables the measurement of angles and lengths.
- -- Orthogonality
- Two vectors are orthogonal if their inner product equals zero, indicating they are at right angles.
- -- Change of Basis
- The process of expressing a vector in terms of a different set of basis vectors.
- -- Dual Space
- The space of all linear functionals on a vector space, crucial for various applications in engineering.
- -- Computational Tools
- Software applications that implement vector space concepts for solving complex engineering tasks.
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