Vector spaces

• What Is the Intuition Behind a Basis in a Vector Space?
  A basis is a nonredundant set of vectors that spans a space, giving every vector a unique set of coordinates relative to that set.
• Difference Between the Dimension and Rank of a Matrix
  Learn why matrix size is not the same as rank, how rank is defined as a space dimension, and how to calculate both with an example.
• How Do You Find the Dimensions of Subspace Sums and Intersections?
  For finite-dimensional subspaces, compute dimensions of sums and intersections with the dimension formula, and see why unions usually fail to be subspaces.
• Do All Linear Transformations Map Zero to Zero?
  Every linear transformation sends the domain's zero vector to the codomain's zero vector, as follows directly from additivity; the reverse need not hold.
• How to Determine Whether a Set Spans ℝ²
  Check whether vectors span ℝ² by testing whether their matrix has rank 2; for two vectors, compute the determinant and check it is nonzero.
• What Is an Example of a Nonstandard Inner Product?
  A weighted dot product such as 2x₁y₁ + 3x₂y₂ is a valid inner product that differs from the standard one in the same coordinates.