Learn & Review: Linear transformations and matrices

Summary of Vector Coordinates, Basis, Linear Combinations, and Span

This summary explains fundamental concepts in linear algebra, focusing on how vectors are represented numerically and the geometric interpretations of operations involving them.

1. Vector Coordinates and Basis Vectors

• Vector Coordinates: A way to represent vectors using numbers. For example, a 2D vector can be represented by a pair of numbers like (3, -2).
• Interpreting Coordinates as Scalars: Each number in a vector's coordinate can be thought of as a scalar that stretches or shrinks a basis vector.
• Standard Basis Vectors (I hat and J hat):
  • I hat: A unit vector pointing right along the x-axis.
  • J hat: A unit vector pointing straight up along the y-axis.
• Vector as a Sum of Scaled Basis Vectors: A vector like (3, -2) can be seen as 3 times I hat plus -2 times J hat. This is the sum of two scaled basis vectors.
• Basis: A set of special vectors (like I hat and J hat) that, when scaled and added, can represent any vector in a given coordinate system. The scalars are the coordinates.
• Flexibility of Basis: It's possible to choose different basis vectors, leading to a different, but still valid, coordinate system. The numerical representation of a vector depends on the chosen basis.

2. Linear Combinations

• Definition: A linear combination of vectors is formed by scaling each vector by a scalar and then adding the results together.
  • For two vectors, v1 and v2, a linear combination is s1 * v1 + s2 * v2, where s1 and s2 are scalars.
• "Linear" Aspect: If one scalar is fixed and the other varies freely, the tip of the resulting vector traces a straight line.

3. Span

• Definition: The span of a set of vectors is the set of all possible vectors that can be reached by forming linear combinations of those vectors.
• Geometric Interpretation:
  • Span of Two Vectors (2D):
    • If the two vectors are not aligned (linearly independent), their span is the entire 2D plane.
    • If the two vectors are aligned (linearly dependent), their span is a single line passing through the origin.
    • If both vectors are zero, their span is just the origin.
  • Span of Two Vectors (3D): If the two vectors are not aligned, their span forms a flat sheet (a plane) cutting through the origin in 3D space.
  • Span of Three Vectors (3D):
    • If the third vector lies within the span of the first two (linearly dependent), the span remains the same flat sheet.
    • If the third vector is not in the span of the first two (linearly independent), their span becomes the entire 3D space.
• Connection to Operations: The span represents all vectors achievable using only vector addition and scalar multiplication.
• Vectors as Points: When dealing with collections of vectors (like a span), it's often convenient to think of them as points (the tips of the vectors) rather than arrows, especially when visualizing entire spaces or lines.

4. Linear Dependence and Independence

• Linearly Dependent: A set of vectors is linearly dependent if at least one vector in the set can be expressed as a linear combination of the others. This means one vector is "redundant" and doesn't add to the span.
• Linearly Independent: A set of vectors is linearly independent if each vector genuinely adds a new "dimension" to the span, meaning no vector can be formed by combining the others.

5. Basis (Technical Definition)

• Definition: A basis of a space is a set of linearly independent vectors that span that entire space.
• Significance: This definition makes sense because linearly independent vectors ensure no redundancy, and spanning the space means all possible vectors within that space can be reached.