# Shared Flashcard Set

## Details

EA Midterm 2
Linear
63
Engineering
11/11/2014

Term
 A system of linear equations is said to be consistent if
Definition
 it has either one solution or infinitely many solutions
Term
 A system of linear equations is said to be inconsistent if
Definition
 it has no solution
Term
 If the augmented matrices of two linear systems of equations are row equivalent, then...
Definition
 the two systems have the same solution set
Term
 A rectangular matrix is in row echelon form if...
Definition
 1. All nonzero rows are above rows of zeros2. Each leading entry of a row is to the right of the leading entry of the row above it3. All entries below a leading entry are zeros
Term
 A rectangular matrix is in reduced row echelon form if...
Definition
 1. All nonzero rows are above rows of zeros2. Each leading entry of a row is to the right of the leading entry of the row above it3. All entries below a leading entry are zeros4. The leading entry in each nonzero row is 15. Each leading entry is the only nonzero entry in its column
Term
 True or False? Each matrix is row equivalent to one and only one reduced echelon matrix
Definition
 True
Term
 A linear system of a equations is consistent if and only if...
Definition
 the rightmost column of the augmented matrix is not a pivot column
Term
 If u and v in R2 are represented as points on a plane, then u + v corresponds to...
Definition
 The fourth vertex of a parallelogram where the other three vertices are 0, u, and v
Term
 Write the following system in Ax = b form.x1 + 2x2 - x3= 4-5x2+ 3x3 = 1
Definition
 x1[1;0] + x2[2;-5] + x3[-1;3] = [4;1][1 2 -1; 0 -5 3][x1; x2; x3] = [4;1]
Term
 If A is an m x n matrix, with columns a1, a2...an, and if b is in Rm, the matrix equation Ax = b has the same solution set as the vector equation
Definition
 x1a1 + x2a2 + ... + xnan = bwhich has the same solution set as the linear system of equations whose augmented matrix is [a1 a2 ... an b]
Term
 The equation Ax = b has a solution if and only if b...
Definition
 is a linear combination of the columns of A.
Term
 Let A be an m x n matrix. The following are logically equivalent (all true or all false)
Definition
 a. For each b in Rm, the equation Ax = b has a solutionb. Each b in Rm is a linear combination of the columns of Ac. The columns of A span Rmd. A has a pivot in every row
Term
 If the product Ax is defined, then the ith entry in Ax is the...
Definition
 sum of the products of corresponding entries from row i of A and from the vector x
Term
 If A is an m x n matrix, and u and v are vectors in Rn, and C is a scalar, then:
Definition
 a. A(u + v) = Au+ Avb. A(cu) = cA(u)
Term
 A linear system of equations is said to be homogenous if it can be written in the form
Definition
 Ax = 0
Term
 The homogenous equation Ax = 0 has a nontrivial solution if and only if
Definition
 the equation has at least one free variable
Term
 Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution of Ax = b is the set of all vectors of the form...
Definition
 w = p + vh, where vh is any solution to the homogenous equation Ax = 0
Term
 An indexed set of vectors {v1, v2...vp} in Rn is said to be linearly independent if
Definition
 the vector equation x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution
Term
 An indexed set of vectors {v1, v2...vp} in Rn is said to be linearly dependent if
Definition
 if there exists weights c1, c2...cp, not all zero, such that c1v1 + c2v2 + ... + cpvp = 0
Term
 The columns of A are linearly independent if and only if
Definition
 the equation Ax = 0 has only the nontrivial solution
Term
 A set of two vectors {v1, v2} is linearly dependent if
Definition
 at least one of the vectors is the multiple of another.
Term
 A set of two vectors {v1, v2} is linearly independent if and only if
Definition
 neither of the vectors is a multiple of the other
Term
 An indexed set S ={v1...vp} of two or more vectors is linearly independent if and only
Definition
 if at least one of the vectors in S is a linear combination of the others
Term
 If a set contains more vectors than there are entries in each vector, then...
Definition
 the set is linearly dependent. {v1...vp} is linearly dependent if p > n
Term
 If a set S = {v1...vp} in Rn contains zero vector, then
Definition
 the set is linearly dependent
Term
 A transformation T from Rn to Rm is a rule that
Definition
 assigns to each vector x in Rn a vector x in Rn a vector T(x) (image of x) in Rm. Rn is the domain. Rm is the codomain.
Term
 Set of all images T(x) is called
Definition
 the range of T
Term
 Transformation T: R2-->R2 defined by T(x) is Ax is called
Definition
 a shear transformation
Term
 A transformation T is linear if...
Definition
 T(u + v) = T(u) + T(v)
Term
 T(cu)= ...
Definition
 cT(u)
Term
 If T is a linear transformation, T(0) =
Definition
 0
Term
 T(c1v1 + ... + cpvp) =...
Definition
 c1T1(v1) + ... cpTp(vp)
Term
 Reflection across x1-axis
Definition
 [1 0; 0 -1]
Term
 Reflection across x2-axis
Definition
 [-1 0; 0 1]
Term
 Reflection across x1 = x2
Definition
 [0 1; 1 0]
Term
 Reflection across x1 = x2
Definition
 [0 1; 1 0]
Term
 Reflection across x1 = -x2
Definition
 [0 -1; -1 0]
Term
 Reflection through origin
Definition
 [-1 0; 0 -1]
Term
 Horizontal expansion/contraction
Definition
 [k 0; 0 1]
Term
 Vertical expansion/contraction
Definition
 [1 0; 0 k]
Term
 Horizontal shear
Definition
 [1 k; 0 1]
Term
 Vertical shear
Definition
 [1 0;k 1]
Term
 Projection onto x1
Definition
 [1 0; 0 0]
Term
 Projection onto x2
Definition
 [0 0; 0 1]
Term
 A mapping T: Rn to Rm is said to be onto Rm if each b in Rm...
Definition
 is the image of at least one x in Rn
Term
 A mapping T: Rn to Rm is said to be one-to-one if each b in Rm...
Definition
 is the image of at most one x in Rn
Term
 Let T: Rn->Rm be a linear transformation. Then T is one-to-one if and only if...
Definition
 the equation T(x) = 0 has only the trivial solution
Term
 Let T: Rn -> Rm be a linear transformation and let A be the standard matrix for T: Then
Definition
 1. T maps Rn onto Rm if and only if the columns of A span Rm2. T is one to one if and only if the columns of A are linearly independent
Term
 Two matrices are equal if and only if
Definition
 1. Same number of rows/columns2. If their corresponding columns are equal
Term
 If A is an m x n matrix, and if B is an n x p matrix with columns b1,...bp, then
Definition
 the product AB is the m x p matrix whose columns are Ab1...AbpAB = A[b1 + b2 +...+ bp] = [Ab1 Ab2 ... Abp]
Term
 (AB)^T =
Definition
 (B^T)(A^T)
Term
 An n x n matrix A is said to be invertible if there is an n x n matrix C such that
Definition
 CA = I and AC = I
Term
 A matrix that is not invertible is called a
Definition
 singular matrix
Term
 Let A = [a b;c d], if A is invertible,
Definition
 ac - bd ~= 0
Term
 det A =
Definition
Term
 If A is an invertible n x n matrix, then for each b in Rn, the equation Ax = b has a unique solution
Definition
 x = (A^-1)b
Term
 If A is an invertible matrix, then
Definition
 A^-1 is invertible and (A^-1)^-1 = A
Term
 If A and B are n x n invertible matrices, then
Definition
 so is AB and (AB)^-1 = (B^-1)(A^-1)
Term
 If A is an invertible matrix, then
Definition
 (A')^-1 = (A^-1)'
Term
 An elementary matrix is
Definition
 obtained by performing a single elementary row operation on an identity matrix
Term
 If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as
Definition
 EA, where the m x m matrix E is created by performing the same row operation on Im (identity in Rm)
Term
 Each elementary matrix E is invertible. The inverse of E is
Definition
 the elementary matrix of the same type that transforms E back into I
Term
 An n x n matrix A is invertible if and only if
Definition
 A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In and In to A^-1
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