Questions and Answers
What is a Linear Equation?
A linear equation in 'n' variables is an equation which can be written in the form a1x1 + a2x2 + ... + anxn = b where a1 is a real number for i = 1, 2, ... , n and so is b.
What is a solution to an equation?
A solution to an equation using 'n' variables is an ordered 'n-tuple' (c1, c2, ... , cn) such that when xi is replaced with ci for all i = 1, 2, ... , n, the resulting statement is true.
What is a system of linear equations?
A system of linear equations is taken to be at least one (probably more) linear equations considered jointly. A solution to a system of linear equations is simply a solution to every equation in the system (if such a thing exists).
What does the Theorem about a System of Linear Equations state?
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What kind of solution(s) will you have in a consistent system?
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What kind of solution(s) does an inconsistent system have?
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What is a matrix?
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What is Scaling in Elementary Row Operation?
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What is Swapping in Elementary Row Operation?
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What is Replacement Operation in Elementary Row Operations?
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What is a row equivalency?
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When is a matrix said to be in Row Echelon Form?
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What additional condition must a matrix satisfy to be in Reduced Row Echelon Form?
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Where do free variables come from?
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When is a system of linear equations consistent with more than one solution?
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What is a vector?
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For two vectors u and v to be considered equal, they must have?
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In general, to add vectors?
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What is scalar multiplication?
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What is denoted R to the nth power (R^n)?
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What are the 8 structural properties of R to the nth power?
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Vector addition is commutative?
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Vector addition is associative?
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A set consisting of a single vector is linearly independent?
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If the columns of the nxn matrix A are linearly independent, and b⊆R^n, then the equation Ax=b has a unique solution?
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Study Notes
Linear Equations
- A linear equation with "n" variables is expressed in the form (a_1x_1 + a_2x_2 + \ldots + a_nx_n = b), where (a_1) to (a_n) are real numbers and (b) is also real.
Solutions to Equations
- An ordered "n-tuple" ((c_1, c_2, \ldots, c_n)) is a solution if substituting (x_i) with (c_i) satisfies the equation.
Systems of Linear Equations
- A system consists of multiple linear equations considered together, with a solution being valid across all equations.
Theorem on Solutions
- Systems of linear equations can have one unique solution, infinitely many solutions, or no solutions at all.
Consistent and Inconsistent Systems
- Consistent systems yield unique or infinitely many solutions, while inconsistent systems result in no solutions.
Matrices
- An (m \times n) matrix is a rectangular array with (m) rows and (n) columns.
Elementary Row Operations
- Scaling: Multiply all entries of a row by a non-zero constant.
- Swap: Exchange the position of two rows.
- Replacement: Multiply a row by a non-zero scalar and add it to another row, replacing that row.
Row Echelon Form
- A matrix is in row echelon form if:
- The leading entry of each row is to the right of the leading entry in the preceding row.
- All entries below each leading entry are zero.
- Any rows of all zeros are at the bottom.
Reduced Row Echelon Form
- Additional conditions for reduced row echelon form include:
- All leading entries are 1.
- All entries above each leading entry are zero.
Free Variables
- Occur in columns of a matrix that do not contain pivot points.
Linear Combinations
- A vector (b) is a linear combination of other vectors if (b = c_1v_1 + c_2v_2 + \ldots + c_pv_p) for some scalars (c_i).
Vectors
- A vector in (R^n) is an (n \times 1) matrix, comprised of a single column.
Vector Operations
- Two vectors are equal if they have the same number of rows and corresponding entries are equal.
- Vector addition requires both vectors to be the same size, adding corresponding entries.
Scalar Multiplication
- Defined for a scalar (c) and a vector (v), resulting in (c \cdot v), which scales each entry of (v).
Span of Vectors
- The span of vectors (v_1, v_2, \ldots, v_p) in (R^n) is the collection of all possible linear combinations of these vectors.
Linear Independence
- A set of vectors is linearly independent if the only solution to their linear combination equaling zero is the trivial solution (all coefficients are zero).
- A set is linearly dependent if at least one vector in the set can be expressed as a linear combination of the others.
Matrix Equations
- For an (m \times n) matrix (A) and vector (x \in R^n), the product (Ax) creates a linear combination of the columns of (A).
Linear Transformations
- A transformation (T) mapping (R^n) to (R^m) can be defined with a matrix (A) such that (T(x) = Ax).
Properties of Linear Transformations
- Transformation is linear if it satisfies:
- (T(u + v) = T(u) + T(v))
- (T(cu) = cT(u)) for any scalar (c).
Theorems on Vector Spaces
- A linear combination (b) is valid if (b) lies within the span of the provided vectors.
- A single vector can be attacked by the linear independence criterion; if it is non-zero, it is considered linearly independent.
Implications of Solutions
- If Ax = b has a solution, then (b) is necessarily in the span of the columns of matrix (A.
- If a set of vectors has more vectors than its dimensionality (more than "n" in a set of nx1 vectors), then the set is linearly dependent.
Conclusion
- Understanding these concepts provides a foundational overview of linear algebra, crucial for solving equations, analyzing vector spaces, and applying transformations effectively.
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Prepare for your Linear Algebra midterm exam with these flashcards that cover key concepts and definitions. Understand linear equations and solutions to equations using variables. Perfect for quick revisions and enhancing your grasp of the subject matter.
