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Linear Algebra > Exam 1 > Flashcards

Exam 1 Flashcards

1
Q

Echelon Form

A

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.

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2
Q

RREF

A

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

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3
Q

Pivot Position

A

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

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4
Q

Pivot Column

A

A column in a matrix that contains a pivot position.

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5
Q

Basic Variables

A

Variables that correspond to a pivot.

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6
Q

Free Variables

A

Variables that do NOT correspond to a pivot.

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7
Q

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

A

It is not consistent.

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8
Q

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

A

It has a unique solution.

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9
Q

If a linear system is consistent and has free variables,

A

It has infinite solutions.

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10
Q

Linear Combination

A

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

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11
Q

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

A

There exist c’s 1&2 s.t. c1v1+c2v2=y.

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12
Q

2 vectors in R2 span R2 if

A

They are not scalar multiples of each other.

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13
Q

Span Rn

A

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

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14
Q

Span

A

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

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15
Q

If A is an mxn matrix, it has

A

m rows and n columns.

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16
Q

If Ax=b has a solution,

A

b is a linear combination of the columns of A.

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17
Q

Homogeneous systems

A

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

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18
Q

Inhomogeneous systems

A

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

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19
Q

If a homogeneous system has a non-trivial solution,

A

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

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20
Q

The Trivial Solution

A

x=0

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21
Q

A set of vectors are linearly independent if

A

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

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22
Q

A set of vectors are linearly dependent if

A

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

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23
Q

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

A

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

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24
Q

2 vectors in Rn are linearly dependent when

A

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

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25
Q

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

A

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

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26
Q

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

A

Rn

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27
Q

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

A

Rm

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28
Q

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

A

The vector T(x).

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29
Q

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

A

The set of all possible vectors T(x).

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30
Q

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

A

Find the span of the columns of A.

31
Q

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

A

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.

32
Q

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

A

Yes, but only in a LINEAR transformation.

33
Q

Standard Vectors

A

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.

34
Q

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

A

The standard matrix for the transformation T.

35
Q

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

A

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

36
Q

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

A

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

37
Q

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

A

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

38
Q

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

A

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

39
Q

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

A

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

40
Q

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

A

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

41
Q

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

A

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

42
Q

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

A

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

43
Q

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

A

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

44
Q

If a linear transformation is onto,

A

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

45
Q

If a linear transformation is one-to-one,

A

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

46
Q

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

A

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

47
Q

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

A

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

48
Q

(A^T)^T=

A

A

49
Q

(A+B)^T=

A

A^T+B^T

50
Q

(rA)^T=

A

r*A^T

51
Q

(AB)^T=

A

B^TA^T

52
Q

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

A

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.

53
Q

Singular

A

NOT Invertible

54
Q

(AB)^-1=

A

B^-1A^-1

55
Q

(A^T)^-1=

A

(A^-1)^T

56
Q

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

A

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.

57
Q

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

A

AX+BY

58
Q

A=LU if

A

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

59
Q

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

A

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

60
Q

How do you use LU to solve Ax=b?

A

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

61
Q

How do you find L in an LU facotrization?

A

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.

62
Q

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

A

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

63
Q

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

A

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

64
Q

A subset of Rn is

A

a collection of vectors in Rn.

65
Q

A subset H of Rn is a subspace if

A

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

66
Q

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

A

The subspace of Rm spanned by a1…an.

67
Q

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

A

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

68
Q

What is the basis for a subspace H?

A

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

69
Q

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,

A

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

70
Q

dim H/the dimension of H

A

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

71
Q

dim(NullA)=

A

The number of free variables in A.

72
Q

dim(colA)=

A

The number of pivot columns in A.

73
Q

RankA=

A

dim(colA).

74
Q

RankA+Dim(NullA)=

A

The number of columns in A.

Linear Algebra (1 decks)

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