Overview
This lecture introduces the fundamental operations of addition and scalar multiplication for vectors and matrices, explains their rules, and provides geometric interpretations.
Fundamental Operations in Linear Algebra
- Three key operations: addition, scalar multiplication, and matrix/vector multiplication.
- Addition and scalar multiplication apply to scalars, vectors, and matrices with simple rules.
- Matrix/vector multiplication is more complex and will be covered separately.
Addition of Vectors and Matrices
- Only objects of the same type and size can be added.
- To add vectors or matrices, add corresponding elements together.
- Example: (0, 1, 2) + (–1, 5, 2) = (–1, 6, 4).
- Example: Matrix addition uses the same position-by-position rule.
- Cannot add objects of different sizes, shapes, or types.
Scalar Multiplication
- Multiply every element of a vector or matrix by the same scalar.
- The resulting object has the same size as the original.
- Example: 7 × (1, –2, 4) = (7, –14, 28).
- Scalar multiplication does not change the type or dimensions.
Linear Combinations
- A linear combination is a sum of scalar multiples of vectors or matrices.
- Example: ( a\vec{x} + b\vec{y} ) is a linear combination of vectors ( \vec{x} ) and ( \vec{y} ).
- Linear combinations only make sense for objects of the same type and size.
- General form: ( \sum_{i=1}^{m} c_i \vec{v}_i ).
Geometric Interpretation
- Vectors can be visualized as points or directed line segments in space.
- Scalar multiplication changes a vector’s length; positive values preserve direction; negative values reverse it.
- Vector addition forms a parallelogram; the resulting vector is the diagonal.
- Subtracting vectors is the same as adding a negative; geometrically, it connects the heads of the two vectors.
Key Terms & Definitions
- Scalar — a single number used to multiply vectors or matrices.
- Vector — an ordered list of numbers, visualized as a point or arrow in space.
- Matrix — a rectangular array of numbers arranged in rows and columns.
- Linear Combination — a sum of scalar multiples of vectors or matrices.
Action Items / Next Steps
- Watch the follow-up video with worked problems on addition and scalar multiplication.
- Prepare for the next lecture on matrix multiplication.