Overview
Linear combinations are a fundamental concept in linear algebra, allowing us to create new vectors from existing ones by scaling and adding them. Understanding linear combinations is essential for exploring vector spaces, which are collections of vectors that can be combined in various ways. The spa...
Key Terms
Example: Velocity is a vector because it has both speed and direction.
Example: In the equation v = 3u, 3 is a scalar.
Example: c1*v1 + c2*v2 is a linear combination of vectors v1 and v2.
Example: The span of {v1, v2} includes all vectors that can be formed from v1 and v2.
Example: The standard basis in R² is {(1,0), (0,1)}.
Example: R³ has a dimension of 3.
Related Topics
Matrix Theory
Study of matrices and their properties, crucial for understanding linear transformations.
intermediateEigenvalues and Eigenvectors
Explore how eigenvalues and eigenvectors relate to linear transformations and vector spaces.
advancedLinear Transformations
Learn about functions that map vectors to vectors, preserving vector addition and scalar multiplication.
intermediateKey Concepts
VectorsScalarsSpanBasis