Questions and Answers
Two matrices are row equivalent if they have the same number of rows.
False
Elementary row operators on an augmented matrix never change the solution set associated with the system.
True
Two equivalent linear systems can have different solution sets.
False
A consistent system of linear equations has one or more solutions.
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Every row operation is reversible.
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A 5x6 matrix has 6 rows.
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The solution set of a linear system involving variables x1...xn is a list of numbers (s1...sn) that makes each equation in the system a true statement when the values s1...sn are substituted for x1...xn respectively.
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Two fundamental questions about a linear system involve existence and uniqueness.
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What is Theorem 1 regarding uniqueness of the Reduced Echelon Form?
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What does Theorem 2 state in relation to existence and uniqueness?
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In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
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The row reduction algorithm applies only to augmented matrices for a linear system.
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A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
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Finding the parametric description of the solution set of a linear system is the same as solving the system.
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One row in an echelon form of an augmented matrix implies that the system is inconsistent.
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What is the Parallelogram Rule for Addition?
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What does Theorem 3 state about the matrix equation Ax=b?
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What does Theorem 4 describe regarding coefficient matrices?
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What does Theorem 5 state about the properties of matrix transformations?
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The equation Ax=b is referred to as the vector equation.
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A vector b is a linear combination of the columns of A if and only if the equation Ax=b has at least one solution.
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The equation Ax=b is consistent if the augmented matrix has a pivot position in every row.
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The first entry in the product Ax is a sum of products.
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If the columns of an mxn matrix A span R^m, then the equation Ax=b is consistent for each b in R^m.
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If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in R^m, then A cannot have a pivot position in every row.
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Suppose Ax=b has a solution. Explain why the solution is unique precisely when Ax=0 has only the trivial solution.
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Let A be a 3x3 matrix with two pivot points: Does the equation Ax=0 have a nontrivial solution?
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Let A be a 3x3 matrix with two pivot points: Does the equation Ax=b have at least one solution for every possible b?
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A homogeneous equation is always consistent.
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The equation Ax=0 gives an explicit description of its solution set.
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The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.
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The equation x=p+tv describes a line through v and parallel to p.
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The solution set of Ax=0 is the set of all vectors of the form w=p+vn where vn is any solution to Ax=0.
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The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution.
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If S is a linearly dependent set, then each vector is a linear combination of the preceding vectors in S.
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The columns of any 4x5 matrix are linearly dependent.
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If x and y are linearly independent and if {x y z} is linearly dependent then z is in the span{xy}.
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If A is an mxn matrix, the columns of A are linearly independent if and only if A has ___ pivot columns.
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How many rows and columns must a matrix A have in order to define a mapping from R^7 into R^8 by the rule T(x)=A(x)?
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A is a 4x7 matrix, what must be a and b in order to define T:R^a -> R^b by T(x)=A(x)?
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If A is a 4x3 matrix then x->Ax maps R^3 onto R^4?
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Every linear transformation from R^n to R^m is a matrix transformation.
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The columns of the standard matrix for a linear transformation from R^n to R^m are the images of the columns of the nxn identity matrix under T.
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A mapping T:R^n-> R^m is said to be one-to-one if each vector in R^n maps onto a unique vector in R^m.
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The standard matrix of horizontal shear transformation of R^2-> R^2 has the form [a 0; 0 d].
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Explain how the columns of A change when A is multiplied by D on the right or on the left.
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Find B such that AB=BA [not the Inxn].
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In order for a matrix B to be the inverse of A both AB= I and BA= I must be true.
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If A and B are nxn and invertible then A^-1 B^-1 is the inverse of AB.
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If A=[a b; c d] and ab-cd/=0 then A is invertible.
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If A is an invertible nxn matrix then the equation Ax=b is consistent for each b in R^n.
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Each elementary matrix is invertible.
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Explain why the columns of an nxn matrix A are linearly independent when A is invertible.
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Suppose A is an nxn matrix and the equation Ax=b has a solution for each b in R^n. Explain why A has to be invertible.
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If the equation Ax=0 has only the trivial solution then A (nxn) is row equivalent to the nxn identity matrix.
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If all the columns of A (nxn) span R^n then the columns are linearly independent.
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If A is an nxn matrix then the equation Ax=b has at least one solution for each b in R^n.
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If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
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If A^T is not invertible then A is not invertible.
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Suppose F is a 5x5 matrix whose column space is not equal to R^5. What can you say about Nul(F)?
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If Q is a 4x4 matrix and Col(Q)=R^4 what can you say for the solutions of the equation Qx=b for b in R^4?
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What does Theorem 6 state about the consistency of Ax=b?
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What does Theorem 7 characterize about linearly dependent sets?
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What does Theorem 8 state regarding a set containing more vectors than entries?
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What does Theorem 9 involve regarding sets that contain the zero vector?
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What does Theorem 10 explain about linear transformations?
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What does Theorem 11 state about one-to-one transformations?
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What does Theorem 12 define in relation to linear transformations and their standard matrices?
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What does the Invertible Matrix Theorem declare regarding equivalence?
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Study Notes
Matrix Equivalence and Operations
- Row equivalence allows transformation between matrices using row operations, not just the same number of rows.
- Elementary row operations maintain the solution set of a linear system, ensuring equivalent systems.
- Two linear systems are equivalent only if they produce the same solution set.
- A consistent system must have at least one solution, while inconsistency implies no solutions.
Solutions in Linear Systems
- Unique solutions occur only when the ax=0 equation has only the trivial solution.
- The existence of free variables indicates infinite solutions in a consistent system.
- Row reduction is applicable to any matrix, not limited to augmented matrices of linear systems.
- Basic variables tie directly to pivot columns and are essential for constructing solutions.
Theorems on Linear Systems
- Reduced echelon forms are unique to each matrix, ensuring consistent results from row reductions.
- Consistency and uniqueness are determined by the structure of the augmented matrix; pivot placements matter.
- The existence and uniqueness theorem relates the existence of solutions to pivot placements in augmented matrices.
Linear Dependence and Independence
- Linearly dependent vectors mean at least one vector is expressible as a combination of others.
- A matrix's columns are guaranteed to be dependent if the number of columns exceeds the number of rows.
- Independence is confirmed if Ax=0 has only the trivial solution, and can be checked via the matrix's pivot structure.
Matrix Representation of Linear Transformations
- Each linear transformation corresponds to a unique matrix capturing the transformation's effects on vector spaces.
- To map R^n to R^m, a matrix with appropriate dimensions (m rows, n columns) is required.
- A transformation is one-to-one only if every input produces a distinct output.
Invertible Matrices and Their Properties
- Invertible matrices ensure consistent solutions for any vector b in R^n, as every b corresponds to a solvable Ax=b equation.
- The invertibility of A is tied to its column space and whether it spans its space, indicated by the number of pivot positions.
- The identity matrix serves as a critical point of reference for determining invertibility and row equivalency.
Characterization Theorems
- Direct relationships exist among linear independence, span, and pivot positions.
- Various theorems characterizing linear systems, from dependence to transformations, highlight crucial properties shared by equivalent statements.
- Certain conditions, such as the presence of a zero vector or reliance on more vectors than dimensions, guarantee linear dependence.
Implications of Solutions and Methods
- The solution set for homogeneous equations provides insights into linear combinations and spans of vectors.
- Parameters in linear systems describe solutions, and transformations change dynamically based on augmentations or adjustments in matrices.
- Studies emphasize the versatility of linear transformations and their applicable forms in various dimensions of input and output spaces.
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Test your knowledge of linear algebra concepts with these flashcards focused on row equivalence and elementary row operations. Perfect for students in MA 265 at Purdue, these cards will help reinforce key definitions and properties essential for understanding matrix operations. Challenge yourself and improve your grasp of fundamental linear algebra topics today!