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Questions and Answers
What is the vector in R^n?
What is the vector in R^n?
What is the trivial solution?
What is the trivial solution?
When the resulting vector, to the solution of a system of equations, has all coordinates equaling 0.
What is the zero vector?
What is the zero vector?
A vector with 0's for every coordinate.
What defines a homogeneous system of linear equations?
What defines a homogeneous system of linear equations?
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What characterizes a non-homogeneous system of linear equations?
What characterizes a non-homogeneous system of linear equations?
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What is an identity matrix?
What is an identity matrix?
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What is scalar multiplication?
What is scalar multiplication?
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What does vector addition involve?
What does vector addition involve?
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What are the components of a vector?
What are the components of a vector?
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What is an additive inverse?
What is an additive inverse?
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What happens when a vector is multiplied by a matrix with a determinant of 0?
What happens when a vector is multiplied by a matrix with a determinant of 0?
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The vector (x, y) in R2 is the same as the vector (x, y, 0) in R3.
The vector (x, y) in R2 is the same as the vector (x, y, 0) in R3.
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Each vector (x, y, z) in R3 has exactly one additive inverse.
Each vector (x, y, z) in R3 has exactly one additive inverse.
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The solution set to a linear system of 4 equations and 6 unknowns consists of a collection of vectors in R6.
The solution set to a linear system of 4 equations and 6 unknowns consists of a collection of vectors in R6.
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For every vector (x1, x2,..., xn) in Rn, the vector (−1) · (x1, x2,..., xn) is an additive inverse.
For every vector (x1, x2,..., xn) in Rn, the vector (−1) · (x1, x2,..., xn) is an additive inverse.
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Study Notes
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.
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Description
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.
