MA 265 Purdue Flashcards
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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.

<p>True</p> Signup and view all the answers

Every row operation is reversible.

<p>True</p> Signup and view all the answers

A 5x6 matrix has 6 rows.

<p>False</p> Signup and view all the answers

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.

<p>False</p> Signup and view all the answers

Two fundamental questions about a linear system involve existence and uniqueness.

<p>True</p> Signup and view all the answers

What is Theorem 1 regarding uniqueness of the Reduced Echelon Form?

<p>Each matrix is row equivalent to one and only one reduced echelon matrix.</p> Signup and view all the answers

What does Theorem 2 state in relation to existence and uniqueness?

<p>A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column.</p> Signup and view all the answers

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

<p>False</p> Signup and view all the answers

The row reduction algorithm applies only to augmented matrices for a linear system.

<p>False</p> Signup and view all the answers

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

<p>True</p> Signup and view all the answers

Finding the parametric description of the solution set of a linear system is the same as solving the system.

<p>False</p> Signup and view all the answers

One row in an echelon form of an augmented matrix implies that the system is inconsistent.

<p>False</p> Signup and view all the answers

What is the Parallelogram Rule for Addition?

<p>If u and v in R^2 are represented as points in the plane, $u+v$ corresponds to the fourth vertex of the parallelogram formed by the other vertices, which are u, 0, and v.</p> Signup and view all the answers

What does Theorem 3 state about the matrix equation Ax=b?

<p>If A is an mxn matrix with columns $a_1...a_n$ and if b is in R^m, the equation Ax=b has the same solution set as the vector equation $x_1a_1+x_2a_2+...+x_n a_n=b$.</p> Signup and view all the answers

What does Theorem 4 describe regarding coefficient matrices?

<p>Theorem 4 states conditions under which for each b in R^m, the equation Ax=b has a solution, which is tied to whether A has a pivot position in every row.</p> Signup and view all the answers

What does Theorem 5 state about the properties of matrix transformations?

<p>Theorems state that for an mxn matrix A, if u and v are vectors in R^n and c is a scalar, the properties $A(u+v)=A(u)+A(v)$ and $A(cu)=cA(u)$ hold.</p> Signup and view all the answers

The equation Ax=b is referred to as the vector equation.

<p>False</p> Signup and view all the answers

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.

<p>True</p> Signup and view all the answers

The equation Ax=b is consistent if the augmented matrix has a pivot position in every row.

<p>False</p> Signup and view all the answers

The first entry in the product Ax is a sum of products.

<p>True</p> Signup and view all the answers

If the columns of an mxn matrix A span R^m, then the equation Ax=b is consistent for each b in R^m.

<p>True</p> Signup and view all the answers

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.

<p>True</p> Signup and view all the answers

Suppose Ax=b has a solution. Explain why the solution is unique precisely when Ax=0 has only the trivial solution.

<p>True</p> Signup and view all the answers

Let A be a 3x3 matrix with two pivot points: Does the equation Ax=0 have a nontrivial solution?

<p>True</p> Signup and view all the answers

Let A be a 3x3 matrix with two pivot points: Does the equation Ax=b have at least one solution for every possible b?

<p>False</p> Signup and view all the answers

A homogeneous equation is always consistent.

<p>True</p> Signup and view all the answers

The equation Ax=0 gives an explicit description of its solution set.

<p>False</p> Signup and view all the answers

The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.

<p>False</p> Signup and view all the answers

The equation x=p+tv describes a line through v and parallel to p.

<p>False</p> Signup and view all the answers

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.

<p>False</p> Signup and view all the answers

The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution.

<p>False</p> Signup and view all the answers

If S is a linearly dependent set, then each vector is a linear combination of the preceding vectors in S.

<p>False</p> Signup and view all the answers

The columns of any 4x5 matrix are linearly dependent.

<p>True</p> Signup and view all the answers

If x and y are linearly independent and if {x y z} is linearly dependent then z is in the span{xy}.

<p>True</p> Signup and view all the answers

If A is an mxn matrix, the columns of A are linearly independent if and only if A has ___ pivot columns.

<p>n</p> Signup and view all the answers

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)?

<p>8 rows and 7 columns</p> Signup and view all the answers

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)?

<p>False</p> Signup and view all the answers

If A is a 4x3 matrix then x->Ax maps R^3 onto R^4?

<p>False</p> Signup and view all the answers

Every linear transformation from R^n to R^m is a matrix transformation.

<p>True</p> Signup and view all the answers

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.

<p>True</p> Signup and view all the answers

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.

<p>False</p> Signup and view all the answers

The standard matrix of horizontal shear transformation of R^2-> R^2 has the form [a 0; 0 d].

<p>False</p> Signup and view all the answers

Explain how the columns of A change when A is multiplied by D on the right or on the left.

<p>Right-multiplication by D multiplies each column of A by the corresponding entry of D, while left-multiplication multiplies each row of A.</p> Signup and view all the answers

Find B such that AB=BA [not the Inxn].

<p>There are infinitely many solutions, any multiple of In will do.</p> Signup and view all the answers

In order for a matrix B to be the inverse of A both AB= I and BA= I must be true.

<p>True</p> Signup and view all the answers

If A and B are nxn and invertible then A^-1 B^-1 is the inverse of AB.

<p>False</p> Signup and view all the answers

If A=[a b; c d] and ab-cd/=0 then A is invertible.

<p>False</p> Signup and view all the answers

If A is an invertible nxn matrix then the equation Ax=b is consistent for each b in R^n.

<p>True</p> Signup and view all the answers

Each elementary matrix is invertible.

<p>True</p> Signup and view all the answers

Explain why the columns of an nxn matrix A are linearly independent when A is invertible.

<p>The unique solution of Ax=0 must be the trivial solution x=0, indicating linear independence.</p> Signup and view all the answers

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.

<p>If Ax=b has a solution for every b, A has pivot positions for each row, indicating that it is row equivalent to the identity matrix.</p> Signup and view all the answers

If the equation Ax=0 has only the trivial solution then A (nxn) is row equivalent to the nxn identity matrix.

<p>True</p> Signup and view all the answers

If all the columns of A (nxn) span R^n then the columns are linearly independent.

<p>True</p> Signup and view all the answers

If A is an nxn matrix then the equation Ax=b has at least one solution for each b in R^n.

<p>False</p> Signup and view all the answers

If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.

<p>True</p> Signup and view all the answers

If A^T is not invertible then A is not invertible.

<p>True</p> Signup and view all the answers

Suppose F is a 5x5 matrix whose column space is not equal to R^5. What can you say about Nul(F)?

<p>If Col(F) ≠ R^5, then F is not invertible and Nul(F) contains a nonzero vector.</p> Signup and view all the answers

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?

<p>If Col(Q) = R^4, Q is invertible and the equation Ax=b has a solution for each b in R^4.</p> Signup and view all the answers

What does Theorem 6 state about the consistency of Ax=b?

<p>If the equation Ax=b is consistent for a given b, then the solution set consists of all vectors of the form w=p+vn, where vn is any solution to Ax=0.</p> Signup and view all the answers

What does Theorem 7 characterize about linearly dependent sets?

<p>An indexed set S of two or more vectors is linearly dependent if at least one of the vectors is a linear combination of the others.</p> Signup and view all the answers

What does Theorem 8 state regarding a set containing more vectors than entries?

<p>If a set contains more vectors than there are entries in each vector, then that set is linearly dependent.</p> Signup and view all the answers

What does Theorem 9 involve regarding sets that contain the zero vector?

<p>If a set contains the zero vector, then the set is linearly dependent.</p> Signup and view all the answers

What does Theorem 10 explain about linear transformations?

<p>Let T be a linear transformation from R^n to R^m, there exists a unique matrix A such that T(x)=A(x) for all x in R^n.</p> Signup and view all the answers

What does Theorem 11 state about one-to-one transformations?

<p>T is one-to-one if and only if T(x)=0 has only the trivial solution.</p> Signup and view all the answers

What does Theorem 12 define in relation to linear transformations and their standard matrices?

<p>A transformation T from R^n to R^m maps R^n onto R^m if and only if the columns of A span R^m.</p> Signup and view all the answers

What does the Invertible Matrix Theorem declare regarding equivalence?

<p>A square matrix A is invertible if it meets all key criteria: being row equivalent to the identity matrix, having n pivot positions, and the equation Ax=0 has only the trivial solution.</p> Signup and view all the answers

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!

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