Questions and Answers
What is a Linear Equation?
Describes a first order line in R^n equations of the form a1x1 + a2x2 + ... + anxn = C, where a1...an are R and x1...xn are variables.
What is a System of Linear Equations?
A collection of 1 or more linear equations to be solved simultaneously.
What is a Solution in the context of linear equations?
A set of real numbers s1...sn such that each equation holds if we set x1 = s1, ..., xn = sn.
What is a Solution Set?
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What is a Consistent System?
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What is an Inconsistent System?
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What are Equivalent Systems?
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How do you solve a basic system?
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What does Interchange refer to in solving systems?
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What does Scaling involve?
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What is Replacement in solving systems?
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What is a Matrix?
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What is an Augmented Matrix?
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What is involved in Solving a matrix?
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What is Echelon Form?
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What is Reduced Echelon Form?
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What is a Pivot in a matrix?
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What is a Pivot Position?
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What is a Basic Variable?
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What is a Free Variable?
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What makes a system solvable?
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What makes a solution consistent?
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What is a Vector?
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What is a Scalar?
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What is Scalar Multiplication?
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What is a Linear Combination of a Vector?
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What are the weights in a Scalar Combination of a Vector?
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What is a Span in linear algebra?
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What is Matrix Multiplication?
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What is the formula for matrix multiplication?
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What are the equivalent statements of a m x n matrix where the vector x is real?
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What does Unique mean in the context of linear equations?
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What is a Homogeneous System?
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What is a Nontrivial solution?
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What is a Trivial solution?
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What is Linear Independence?
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What is Linear Dependence?
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What are some facts about Linear Dependence?
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What is a Linear Transformation?
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What is the Domain of Transformation?
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What is the Co-Domain of Transformation?
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What is the Range of Transformation?
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What makes a transformation linear?
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What does Onto mean in the context of transformations?
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What does One-to-One mean in transformations?
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What is Matrix Multiplication?
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What are the Properties of Matrix Multiplication?
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What is a Power Matrix?
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What is Transposition?
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What are the Properties of Transposition?
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What is an Invertible Matrix?
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What is a Singular Matrix?
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What makes a matrix invertible?
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What are the Properties of an Invertible Matrix?
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How do you find an inverse?
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What is an Elementary Matrix?
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What are determinants?
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For what types of matrices are determinants defined?
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What does it mean if the determinant is non-zero?
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What if the matrix has a 0 row or 0 column?
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What is the Cofactor of A?
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What is Cofactor Expansion?
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What is an Upper Triangular Matrix?
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What is a Lower Triangular Matrix?
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How do you find the determinant of a triangular matrix?
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What are the effects of each row operation on the determinant?
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How do you compute the determinant with a triangular matrix?
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What are three random facts about determinants?
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Study Notes
Linear Equations and Systems
- Linear equations represent first-order lines in R^n, following the form a1x1 + a2x2 + ... + anxn = C, where a1...an are real numbers and x1...xn are variables.
- A system of linear equations consists of two or more linear equations that need to be solved simultaneously.
- A solution is a set of real numbers s1, s2, ..., sn that satisfies all equations when substituted for the variables.
- The solution set includes all possible solutions to a system of equations.
System Consistency
- A consistent system has at least one solution.
- An inconsistent system has no solutions.
- Equivalent systems share the same solution set.
Solving Systems
- Basic methods include interchanging, scaling, and replacement to isolate variables.
- Interchanging refers to reordering equations; scaling involves multiplying equations by constants; replacement replaces an equation with a multiple of another.
Matrices
- A matrix (m x n) is a rectangular array of numbers with m rows and n columns.
- An augmented matrix combines the coefficient matrix with the constant solutions.
- To solve a matrix, pivot positions are established; each pivot should equal one, with no other entry in its column.
Matrix Forms
- Echelon Form has all zero rows at the bottom and leading nonzero entries arranged progressively rightwards, with zeroes below each leading entry.
- Reduced Echelon Form maintains the properties of Echelon Form plus requiring each leading entry to be 1 and the only nonzero entry in its column.
Variables and Pivots
- A pivot is the leading nonzero entry in an echelon matrix, while its position is called the pivot position.
- Basic variables correspond to columns with pivots; free variables correspond to columns without pivots and can take any value.
Solution Criteria
- A system is solvable if its echelon form does not contain a row in the format [0...0 C] where C is nonzero.
- A unique solution occurs when there are no free variables and each variable is basic.
Vectors and Scalars
- A vector consists of quantities with direction and magnitude, while a scalar contains only magnitude.
- Scalar multiplication combines a vector with a scalar, affecting the vector's direction based on the scalar's sign.
Span and Transformations
- The span of vectors {v1...vk} is the collection of all possible linear combinations of these vectors.
- Linear transformations, T: R^n -> R^m, map vectors from one space to another.
Properties of Transformations
- A linear transformation satisfies the following: T(u + v) = T(u) + T(v) and T(cu) = cT(u).
- A transformation is onto if its image covers all of R^m; it's one-to-one if distinct inputs yield distinct outputs.
Matrix Operations
- Matrix multiplication involves combining mxn matrix A with nxk matrix B to form a product matrix.
- Properties of matrix multiplication include associativity and distributive properties, while it is not commutative.
Inverses and Determinants
- An invertible matrix A has a corresponding inverse B such that AB = BA = In (identity matrix).
- Determinants are defined for square matrices, indicating whether a matrix is invertible (nonzero determinant) or singular (zero determinant).
Special Matrix Types
- Triangular matrices are categorized into upper and lower triangular forms.
- The determinant of a triangular matrix equals the product of its diagonal elements.
Cofactor and Expansion
- The cofactor of a matrix A is defined as (-1)^(i+j) Aij.
- Cofactor expansion applies to calculate determinants.
Properties of Determinants
- Determinant properties include det(AB) = det(A)det(B) and det(A^T) = det(A).
- Row operations affect determinants: replacement has no effect, while row exchanges flip the sign and scalar multiplication modifies the determinant by that scalar factor.
General Facts
- The zero vector is considered dependent, and if the collection includes the zero vector or k > n, it indicates linear dependence.
- Linear independence of a vector collection exists if only the trivial solution can be formed from their combination.
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Test your knowledge of key concepts in Linear Algebra with these flashcards from Chapter 1. Each card defines important terms like Linear Equations and Systems of Linear Equations that are fundamental to the subject. Perfect for quick revision and learning!