Vector Spaces Flashcards (Linear Algebra)

Vector in R^n

• Represents all ordered n-tuples of real numbers.

Trivial Solution

• Occurs when the solution to a system of equations results in the zero vector, characterized by all coordinates equal to 0.

Zero Vector

• A vector consisting entirely of zeros for every coordinate, unique to each dimension (e.g., (0,0) for 2D, (0,0,0) for 3D, and (0,...,0) for n dimensions).

Homogeneous System of Linear Equations

• Defined by all constant terms being 0, indicating that the equation AX = B has B as the zero vector.

Non-homogeneous System of Linear Equations

• A system characterized by having a unique non-trivial solution, distinct from homogeneous systems.

Identity Matrix

• Contains 1's on the main diagonal and 0's elsewhere. It does not alter a vector when multiplied and is involved in defining matrix inverses.

Scalar Multiplication

• The process of multiplying a matrix by a constant to scale its components.

Vector Addition

• The operation of summing corresponding components from two vectors to produce a resultant vector.

Components

• Refers to the individual coordinates that make up a vector.

Additive Inverse

• A vector that, when added to the original vector, results in the zero vector; for a vector [x,y,z], the additive inverse is [-x,-y,-z].

Matrix Multiplication Impact with Determinant of 0

• Multiplying by a matrix with a determinant of 0 collapses the vector’s dimensions; it can change a 2D shape into a line, effectively losing one dimension.

TRUE OR FALSE: Vector Equivalence in Different Dimensions

• False: The vector (x, y) in R2 is not the same as (x, y, 0) in R3.

TRUE OR FALSE: Existence of Additive Inverse in R3

• True: Each vector (x, y, z) has a unique additive inverse represented as [-x, -y, -z].

TRUE OR FALSE: Solution Set in R6 from Linear Equations

• True: The solution set for a linear system of 4 equations with 6 unknowns is a collection of vectors in R6, noted for the presence of free variables.

TRUE OR FALSE: Additive Inverse Definition in Rn

• True: For any vector (x1, x2,..., xn) in Rn, the vector (-1) · (x1, x2,..., xn) represents its additive inverse.

Test your knowledge of vector spaces in Linear Algebra with these flashcards. Each card covers key concepts and definitions, such as vectors in R^n and the zero vector. Enhance your understanding of foundational topics in this essential math discipline.