Questions and Answers
When is a matrix in echelon form? (Select all that apply)
What does RREF stand for?
Reduced Row Echelon Form
What is a pivot column?
A pivot position is a location that corresponds to a leading 1 in the reduced echelon form.
Define basic variable and free variable.
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What is Span{v1...vp}?
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What is the definition of AX?
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What does it mean for vectors to be linearly independent?
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What is linear dependence?
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What are the conditions for a transformation to be linear?
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When is a linear transformation T: R^n to R^m onto?
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What is a linear transformation one-to-one?
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State the row-column rule for computing AB.
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What does it mean for a linear transformation T: R^n to R^n to be invertible?
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What is an elementary matrix?
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What is the transpose of matrix A?
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What is the significance of Theorem 4?
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When does the homogeneous equation Ax = 0 have a nontrivial solution?
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What does it mean for T to be one-to-one and onto?
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State the Inverse Matrix Theorem.
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If A is an invertible nXn matrix, what can be said about the solutions for each b in R^n?
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Study Notes
Echelon Form
- A matrix is in echelon form if all nonzero rows are positioned above rows of all zeros.
- Each leading entry in a row must be to the right of the leading entry in the row above.
- All entries below a leading entry must be zeros.
Reduced Row Echelon Form (RREF)
- Leading entries in each nonzero row are equal to 1.
- Each leading 1 is the only nonzero entry in its respective column.
Pivot Columns
- A column containing a leading 1 in the reduced echelon form, indicating a pivot position.
Basic and Free Variables
- Basic variables correspond to pivot columns in a matrix.
- Free variables are those that do not correspond to a pivot column.
Span
- Span{v1...vp} represents the set of all linear combinations of vectors v1 to vp in R^n, expressed as c1v1 + c2v2 + ... + cpvp.
Product of Matrices (AX)
- The product of matrix A and vector x is a linear combination of the columns of A, with corresponding entries of x serving as weights.
Linear Independence
- A set of vectors is linearly independent if the equation x1v1 + ... + xpvp = 0 has only the trivial solution.
- No vector in the set can be expressed as a linear combination of the others.
Linear Dependence
- A set of vectors is linearly dependent if there exists a non-trivial combination that equals zero.
- A zero vector in the set indicates linear dependence, as does having more variables than equations.
Linear Transformations
- A transformation T is linear if it satisfies T(U+V) = T(U) + T(V) and T(cU) = cT(U).
- A linear transformation from R^n to R^m is onto if every b in R^m can be produced from at least one x in R^n.
- A transformation is one-to-one if it maps each b in R^m to at most one x in R^n, ensuring no free variables and linear independence of columns.
Row-Column Rule for Matrix Multiplication
- If AB is defined, the entry in row i and column j is the sum of products of row i of A and column j of B.
Invertibility of a Linear Transformation
- A transformation T: R^n to R^n is invertible if there exists an inverse function S such that S(T(x)) = x and T(S(x)) = x.
Elementary Matrices
- Elementary matrices are obtained by performing a single elementary row operation on an identity matrix.
Transpose of a Matrix
- The transpose of an mxn matrix A, denoted A^T, is an nxm matrix, where rows of A become columns of A^T.
Homogeneous Equations
- The homogeneous equation Ax = 0 has a nontrivial solution if at least one free variable exists.
Linear Transformation Conditions
- A linear transformation T maps R^n onto R^m if the columns of its standard matrix A span R^m.
- T is one-to-one if the columns of A are linearly independent.
Inverse Matrix Theorem
- A matrix A is invertible if it is row equivalent to the identity matrix, has n pivot positions, and the equation Ax = 0 has only the trivial solution.
- Its columns form a linearly independent set and span R^n.
Inversion and Linear Transformations
- For an invertible n x n matrix A, there is a unique solution for each b in R^n, maintaining consistency across transformations.
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Test your understanding of matrix algebra with these flashcards. Cover key concepts such as echelon form and reduced row echelon form (RREF). Perfect for anyone studying linear algebra.