Exam 1 Flashcards
Echelon Form
All zero rows are at the bottom,
The first non-zero entry of each row is to the right of any leading entries in the row above it,
All entries below a leading entry are zero.
RREF
Echelon form, plus all leading entries are one and leading entries are the only non-zero entry in their columns.
Pivot Position
A matrix position that corresponds to a leading 1 in the RREF of the matrix.
Pivot Column
A column in a matrix that contains a pivot position.
Basic Variables
Variables that correspond to a pivot.
Free Variables
Variables that do NOT correspond to a pivot.
If an augmented matrix of form A|b has a pivot in the last column,
It is not consistent.
If a linear system is consistent and has no free variables,
It has a unique solution.
If a linear system is consistent and has free variables,
It has infinite solutions.
Linear Combination
A vector resulting from adding multiples of a set of vectors.
y can be represented as a linear combination of v1 and v2 if
There exist c’s 1&2 s.t. c1v1+c2v2=y.
2 vectors in R2 span R2 if
They are not scalar multiples of each other.
Span Rn
Any vector in Rn can be represented as a linear combination of the set of vectors.
Span
The set of all linear combinations of a set of vectors with the same number of real entries.
If A is an mxn matrix, it has
m rows and n columns.
If Ax=b has a solution,
b is a linear combination of the columns of A.
Homogeneous systems
Systems of the form Ax=0 with the trivial solution.
Inhomogeneous systems
Systems of the form Ax=b where b doesn’t equal 0.
If a homogeneous system has a non-trivial solution,
There must be a free variable/A must have a column with no pivot.
The Trivial Solution
x=0
A set of vectors are linearly independent if
There is NO set of constants including non-zero constants that allow the set of vectors’ span to include the zero vector.
A set of vectors are linearly dependent if
There is a set of constants including non-zero constants that allow the set of vectors’ span to include the zero vector.
What solutions to c1v1+c2v2=0 make the set of vectors {v1,v2} linearly dependent?
c1 does not equal zero OR c2 does not equal zero OR neither c1 nor c2 equals zero.
2 vectors in Rn are linearly dependent when
One or both of the vectors are the zero vector OR One vector is a multiple of the other OR both.
If every column in a matrix of the form A=(v1 v2 … vk) is not pivotal,
The set of vectors (v1 v2 … vk) is linearly dependent.
In the function defined T:Rn->Rm, T(x)=Ax, what is the domain of T?
Rn
In the function defined T:Rn->Rm, T(x)=Ax, what is the codomain of T?
Rm
In the function defined T:Rn->Rm, T(x)=Ax, what is the image of x under T?
The vector T(x).
In the function defined T:Rn->Rm, T(x)=Ax, what is the range of T?
The set of all possible vectors T(x).
In the function defined T:Rn->Rm, T(x)=Ax, how do you find the range of T?
Find the span of the columns of A.
A transformation of the form T:Rn->Rm is linear if
T(u+v)=T(u)+T(v) for all u’s and v’s in Rn, and T(cv)=cT(v) for all v’s in Rn and all c’s in R.
In a linear transformation, does T(c1v1+c2v2)=c1T(v1)+c2T(v2)?
Yes, but only in a LINEAR transformation.
Standard Vectors
The vectors e1, … en where e1=a vector with all zeroes except the first entry, which is one, and en= a vector with all zeroes except the nth entry, which is one.
In the transformation T(x)=Ax, if A is a matrix of the form A=(T(e1) T(e2) T(e3)), A is
The standard matrix for the transformation T.
Standard 2x2 matrix for the transformation that rotates vectors counter clockwise by theta?
top left: cos(theta) top right -sin(theta)
bottom left: sin(theta) bottom right cos(theta)
Standard 2x2 matrix for the transformation that reflects vectors through the x1 axis?
top left:1 top right: 0
bottom left: 0 bottom right:-1
Standard 2x2 matrix for the transformation that reflects vectors through the x2 axis?
top left:-1 top right: 0
bottom left: 0 bottom right:1
Standard 2x2 matrix for the transformation that reflects vectors through the line x1=x2?
top left: 0 top right: 1
bottom left:1 bottom right: 0
Standard 2x2 matrix for the transformation that reflects vectors through the line x1=-x2?
top left: 0 top right:-1
bottom left:-1 bottom right: 0
Standard 2x2 matrix for the transformation that performs a horizontal contraction/expansion?
top left: k top right: 0
bottom left: 0 bottom right:1
Standard 2x2 matrix for the transformation that performs a vertical contraction/expansion?
top left:1 top right: 0
bottom left: 0 bottom right: k
Standard 2x2 matrix for the transformation that performs a projection onto the x1 axis?
top left:1 top right: 0
bottom left: 0 bottom right: 0
Standard 2x2 matrix for the transformation that performs a projection onto the x2 axis?
top left: 0 top right: 0
bottom left: 0 bottom right: 1
If a linear transformation is onto,
For any b in Rm, Ax=b has a solution, and A has a pivot in every row.
If a linear transformation is one-to-one,
Ax=0 has only one solution: x=0, and every column of A is pivotal.
For a linear transformation of form T:Rn->Rm with standard matrix A, list the equivalent statements to “T is onto”
A has columns that span Rm and every row of A is pivotal.
For a linear transformation of form T:Rn->Rm with standard matrix A, list the equivalent statements to “T is one-to-one”
The only solution to T(x)=0 is trivial, A has linearly independent columns, and each column of A is pivotal.
(A^T)^T=
A
(A+B)^T=
A^T+B^T
(rA)^T=
r*A^T
(AB)^T=
B^TA^T
How do you find A^-1 if A is an nxn matrix?
Row reduce (A|In) to RREF, if it then has the form (In|B), A is invertible and B=A^-1, otherwise, A is singular.
Singular
NOT Invertible
(AB)^-1=
B^-1A^-1
(A^T)^-1=
(A^-1)^T
If A is an nxn matrix, list the statements equivalent to “A is invertible”
A is row equivalent to In,
A has n pivotal columns,
Ax=0 has only the trivial solution,
The columns of A are linearly independent,
The equation Ax=b has a solution for all b’s in Rn,
The columns of A span Rn,
A has a left and a right inverse,
A^T is invertible,
The columns of A are a basis for Rn,
ColA=Rn,
RankA=dim(ColA)=n, and
NullA=0.
What is the product of the matrix multiplication (A B)(X)
_________________________________________________________(Y)
AX+BY
A=LU if
A is an mxn matrix that can be reduced to RREF without row exchanges.
In the form A=LU, where A is an mxn matrix, L and U are,
L is a lower triangular mxm matrix with ones on the diagonal, U is an echelon form of A.
How do you use LU to solve Ax=b?
Construct the LU decomposition of A, set Ux=y, solve for y in Ly=b, then solve for x in Ux=y.
How do you find L in an LU facotrization?
Find U, multiply the number of each row added to each row by -1, then place the result in the position of L corresponding to x,y, where x is the row that was added and y is the row that was added to.
What is the formula for the Leontif Input-Output model?
(I-C)x=d, where x gives the quantities that meet total demand, C gives internal demands, and d gives external demands.
How can a translation of the form (x,y)->(x+h,y+k) be represented using homogenous coordinates?
A matrix like I3, but replace the top right with h and the middle right with k. Continue similarly for more dimensions.
A subset of Rn is
a collection of vectors in Rn.
A subset H of Rn is a subspace if
For any c in R and for u and v in H cu is in H and u+v is in H.
If A is an mxn matrix of the form (a1 a2 … an), what is the column space of A?
The subspace of Rm spanned by a1…an.
If A is an mxn matrix of the form (a1 a2 … an), what is the null space of A?
The subspace of Rn spanned by the set of vectors x that solve Ax=0.
What is the basis for a subspace H?
A set of linearly independent vectors in H that span H.
If B=b1, b2, … bn is a basis for subspace H, and x is a vector in H, the coordinates of x relative B are,
The weights of the vectors in B s.t. x=c1b1+c2b2+…+cnbn. Also written [x]subB
dim H/the dimension of H
The number of vectors in a basis of the subspace H.
dim(NullA)=
The number of free variables in A.
dim(colA)=
The number of pivot columns in A.
RankA=
dim(colA).
RankA+Dim(NullA)=
The number of columns in A.
