Exam 1

Question 1

Echelon Form

Answer 1

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.

Question 2

RREF

Answer 2

Echelon form, plus all leading entries are one and leading entries are the only non-zero entry in their columns.

Question 3

Pivot Position

Answer 3

A matrix position that corresponds to a leading 1 in the RREF of the matrix.

Question 4

Pivot Column

Answer 4

A column in a matrix that contains a pivot position.

Question 5

Basic Variables

Answer 5

Variables that correspond to a pivot.

Question 6

Free Variables

Answer 6

Variables that do NOT correspond to a pivot.

Question 7

If an augmented matrix of form A|b has a pivot in the last column,

Answer 7

It is not consistent.

Question 8

If a linear system is consistent and has no free variables,

Answer 8

It has a unique solution.

Question 9

If a linear system is consistent and has free variables,

Answer 9

It has infinite solutions.

Question 10

Linear Combination

Answer 10

A vector resulting from adding multiples of a set of vectors.

Question 11

y can be represented as a linear combination of v1 and v2 if

Answer 11

There exist c’s 1&2 s.t. c1v1+c2v2=y.

Question 12

2 vectors in R2 span R2 if

Answer 12

They are not scalar multiples of each other.

Question 13

Span Rn

Answer 13

Any vector in Rn can be represented as a linear combination of the set of vectors.

Question 14

Span

Answer 14

The set of all linear combinations of a set of vectors with the same number of real entries.

Question 15

If A is an mxn matrix, it has

Answer 15

m rows and n columns.

Question 16

If Ax=b has a solution,

Answer 16

b is a linear combination of the columns of A.

Question 17

Homogeneous systems

Answer 17

Systems of the form Ax=0 with the trivial solution.

Question 18

Inhomogeneous systems

Answer 18

Systems of the form Ax=b where b doesn’t equal 0.

Question 19

If a homogeneous system has a non-trivial solution,

Answer 19

There must be a free variable/A must have a column with no pivot.

Question 20

The Trivial Solution

Answer 20

x=0

Question 21

A set of vectors are linearly independent if

Answer 21

There is NO set of constants including non-zero constants that allow the set of vectors’ span to include the zero vector.

Question 22

A set of vectors are linearly dependent if

Answer 22

There is a set of constants including non-zero constants that allow the set of vectors’ span to include the zero vector.

Question 23

What solutions to c1v1+c2v2=0 make the set of vectors {v1,v2} linearly dependent?

Answer 23

c1 does not equal zero OR c2 does not equal zero OR neither c1 nor c2 equals zero.

Question 24

2 vectors in Rn are linearly dependent when

Answer 24

One or both of the vectors are the zero vector OR One vector is a multiple of the other OR both.

Question 25

If every column in a matrix of the form A=(v1 v2 ... vk) is not pivotal,

Answer 25

The set of vectors (v1 v2 ... vk) is linearly dependent.

Question 26

In the function defined T:Rn->Rm, T(x)=Ax, what is the domain of T?

Answer 26

Rn

Question 27

In the function defined T:Rn->Rm, T(x)=Ax, what is the codomain of T?

Answer 27

Rm

Question 28

In the function defined T:Rn->Rm, T(x)=Ax, what is the image of x under T?

Answer 28

The vector T(x).

Question 29

In the function defined T:Rn->Rm, T(x)=Ax, what is the range of T?

Answer 29

The set of all possible vectors T(x).

Question 30

In the function defined T:Rn->Rm, T(x)=Ax, how do you find the range of T?

Answer 30

Find the span of the columns of A.

Question 31

A transformation of the form T:Rn->Rm is linear if

Answer 31

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.

Question 32

In a linear transformation, does T(c1v1+c2v2)=c1T(v1)+c2T(v2)?

Answer 32

Yes, but only in a LINEAR transformation.

Question 33

Standard Vectors

Answer 33

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.

Question 34

In the transformation T(x)=Ax, if A is a matrix of the form A=(T(e1) T(e2) T(e3)), A is

Answer 34

The standard matrix for the transformation T.

Question 35

Standard 2x2 matrix for the transformation that rotates vectors counter clockwise by theta?

Answer 35

top left: cos(theta) top right -sin(theta) bottom left: sin(theta) bottom right cos(theta)

Question 36

Standard 2x2 matrix for the transformation that reflects vectors through the x1 axis?

Answer 36

top left:1 top right: 0 bottom left: 0 bottom right:-1

Question 37

Standard 2x2 matrix for the transformation that reflects vectors through the x2 axis?

Answer 37

top left:-1 top right: 0 bottom left: 0 bottom right:1

Question 38

Standard 2x2 matrix for the transformation that reflects vectors through the line x1=x2?

Answer 38

top left: 0 top right: 1 bottom left:1 bottom right: 0

Question 39

Standard 2x2 matrix for the transformation that reflects vectors through the line x1=-x2?

Answer 39

top left: 0 top right:-1 bottom left:-1 bottom right: 0

Question 40

Standard 2x2 matrix for the transformation that performs a horizontal contraction/expansion?

Answer 40

top left: k top right: 0 bottom left: 0 bottom right:1

Question 41

Standard 2x2 matrix for the transformation that performs a vertical contraction/expansion?

Answer 41

top left:1 top right: 0 bottom left: 0 bottom right: k

Question 42

Standard 2x2 matrix for the transformation that performs a projection onto the x1 axis?

Answer 42

top left:1 top right: 0 bottom left: 0 bottom right: 0

Question 43

Standard 2x2 matrix for the transformation that performs a projection onto the x2 axis?

Answer 43

top left: 0 top right: 0 bottom left: 0 bottom right: 1

Question 44

If a linear transformation is onto,

Answer 44

For any b in Rm, Ax=b has a solution, and A has a pivot in every row.

Question 45

If a linear transformation is one-to-one,

Answer 45

Ax=0 has only one solution: x=0, and every column of A is pivotal.

Question 46

For a linear transformation of form T:Rn->Rm with standard matrix A, list the equivalent statements to "T is onto"

Answer 46

A has columns that span Rm and every row of A is pivotal.

Question 47

For a linear transformation of form T:Rn->Rm with standard matrix A, list the equivalent statements to "T is one-to-one"

Answer 47

The only solution to T(x)=0 is trivial, A has linearly independent columns, and each column of A is pivotal.

Question 48

(A^T)^T=

Answer 48

A

Question 49

(A+B)^T=

Answer 49

A^T+B^T

Question 50

(rA)^T=

Answer 50

r*A^T

Question 51

(AB)^T=

Answer 51

B^TA^T

Question 52

How do you find A^-1 if A is an nxn matrix?

Answer 52

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.

Question 53

Singular

Answer 53

NOT Invertible

Question 54

(AB)^-1=

Answer 54

B^-1A^-1

Question 55

(A^T)^-1=

Answer 55

(A^-1)^T

Question 56

If A is an nxn matrix, list the statements equivalent to "A is invertible"

Answer 56

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.

Question 57

What is the product of the matrix multiplication (A B)(X) _________________________________________________________(Y)

Answer 57

AX+BY

Question 58

A=LU if

Answer 58

A is an mxn matrix that can be reduced to RREF without row exchanges.

Question 59

In the form A=LU, where A is an mxn matrix, L and U are,

Answer 59

L is a lower triangular mxm matrix with ones on the diagonal, U is an echelon form of A.

Question 60

How do you use LU to solve Ax=b?

Answer 60

Construct the LU decomposition of A, set Ux=y, solve for y in Ly=b, then solve for x in Ux=y.

Question 61

How do you find L in an LU facotrization?

Answer 61

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.

Question 62

What is the formula for the Leontif Input-Output model?

Answer 62

(I-C)x=d, where x gives the quantities that meet total demand, C gives internal demands, and d gives external demands.

Question 63

How can a translation of the form (x,y)->(x+h,y+k) be represented using homogenous coordinates?

Answer 63

A matrix like I3, but replace the top right with h and the middle right with k. Continue similarly for more dimensions.

Question 64

A subset of Rn is

Answer 64

a collection of vectors in Rn.

Question 65

A subset H of Rn is a subspace if

Answer 65

For any c in R and for u and v in H cu is in H and u+v is in H.

Question 66

If A is an mxn matrix of the form (a1 a2 ... an), what is the column space of A?

Answer 66

The subspace of Rm spanned by a1...an.

Question 67

If A is an mxn matrix of the form (a1 a2 ... an), what is the null space of A?

Answer 67

The subspace of Rn spanned by the set of vectors x that solve Ax=0.

Question 68

What is the basis for a subspace H?

Answer 68

A set of linearly independent vectors in H that span H.

Question 69

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,

Answer 69

The weights of the vectors in B s.t. x=c1b1+c2b2+...+cnbn. Also written [x]subB

Question 70

dim H/the dimension of H

Answer 70

The number of vectors in a basis of the subspace H.

Question 71

dim(NullA)=

Answer 71

The number of free variables in A.

Question 72

dim(colA)=

Answer 72

The number of pivot columns in A.

Question 73

RankA=

Answer 73

dim(colA).

Question 74

RankA+Dim(NullA)=

Answer 74

The number of columns in A.