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EA Midterm 2
Linear
63
Engineering
Undergraduate 1
11/11/2014
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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 zeros
2. Each leading entry of a row is to the right of the leading entry of the row above it
3. 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 zeros
2. Each leading entry of a row is to the right of the leading entry of the row above it
3. All entries below a leading entry are zeros
4. The leading entry in each nonzero row is 1
5. 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 = b

which 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 solution
b. Each b in Rm is a linear combination of the columns of A
c. The columns of A span Rm
d. 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+ Av
b. 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 Rm
2. 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/columns
2. 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...Abp

AB = 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
ad - bc
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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