Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Linear Algebra > Midterm > Flashcards

Midterm Flashcards

1
Q

Consistant

A

Has more than 0 solutions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Uniqueness

A

Has 1 solution

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Define m and n for Amxn

A

m is rows, n is columns

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

True or false. Each matrix has a unique RREF. As such, two matrices cannot be wrote equivalent to the same RREF

A

False. A given matrix in RREF can be equivalent to an infinite amount of matrices, whilst any given matrix can only have one RREF

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

True or false. An RREF is an REF.

A

True. The rules for an RREF satisfy all conditions for an REF

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Basic Variables

A

Variables which are not free

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

True or false. Rough reduction to the REF is enough to determine if a solution exists and is unique.

A

True, although you may not know the exact solution.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

True or false. The span of any set of vectors is at least a line.

A

Fase. Consider the case of a set that contains only the zero vector.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What is the name of the equation Ax= b

A

The matrix equation

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

What type of matrix is A in the matrix equation Ax =b

A

Coefficient matrix

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Identify the vectors in the homogenous equation Ax=0

A

X and the 0 vector

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

T or F. For any given matrix A, the homogeneous equation has 1 solution.

A

False, for any given matrix A the homogeneous equation has at least one solution (the trivial solution)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What is the geometric interpretation of the following parametric vector equations?
X=tv
X=su+tv

A

Line, plane

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

The solution set for Ax=0 is a line. What can we say about the solution set for Ax=b

A

It is either a line or nonexistent

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Write the following solution set in parametric vector form and determine its geometric interpretation.
X1= 3+4x3
X2=5+7x3
X3 is free

A

[3;5;0]=x3 [4,7,1]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

T or F. A set of three vectors in R3 may or may not be linearly dependent and spans at minimum a line

A

False. They may all be the zero vector

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

If Ax =0 has the trivial solution what is the relation of the columns of A?

A

We cannot say because Ax=0 always has the trivial solution

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

True or False. The range and codomain of a transformation are interchangeable terms that mean the same thing

A

False. The codomain is the space of the transformations of the columns of A. It is possible that the range is the whole codomain, however it is not necessarily true in all scenarios

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

In the transformation given by Ax=b the equation is consistent, identify wether the columns of A and the vectors x and b exist in the range codomain and domain respectively.

A

The columns of A exist in the range (within the codomain), the vector x exists in the domain and the vector b exists in the range (within the codomain)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What is the name of a matrix that represents a linear transformation T?

A

The standard matrix

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

T or F A transformation of Rn to Rm is said to be one to one if each b in Rn has only one solution

A

False. Each b in Rm can have a max of one solution, it can have no solutions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

T or F. A transformation from Rn to Rm is said to be onto if the range is the codomain

A

True

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Identify with which fundamental questions of linear algebra the term onto and one to one correspond with respectively

A

Onto->existence, one to one->uniqueness

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

T(e1) = (1,3,6), T(e2) = (2,8,0) Find the standard matrix domain and codomain of the transformation, and give the geometric description of the range

A

Standard matrix A =[1 2; 3 8; 6 0] domain is R2 and codomain is R3 the range is a plane in R3

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

The transformation T of R2 -> R2 first reflects points through the horizontal x axis then reflects points through the origin. What is the standard matrix of the transformation?

A

A = [-1 0; 01]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

Rotate the vector [2;3] 90 degrees ccw and reflect through the horizontal x axis.

A

[-3;-2]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

List the diagonal entries if matrix A [123;456;789]

A

1,5,9

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

True of False Aand B commute. This the addition A+B is defined

A

Fase. Consider the case where A is 3x4 and B is 4x3

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
29
Q

True or False. The matrix multiplication A(At) is always defined

A

True. If A is mxn, At is nxm

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

True or False. Given a matrix A, At is always defined

A

True

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
31
Q

What terms refer ti a non invertible and invertible matrix respectively

A

Singular and non singular

32
Q

True or False A is am mxn matrix. A is invertible.

A

False. A must be an nxn matrix in order to be invertible

33
Q

Are all nxn matrixes invertible

A

No

34
Q

What is the determinant of the matrix A = [12;-1-2]

A

0

35
Q

Is [12;-1-2] invertible

A

No

36
Q

What are three properties of a subspace

A

It must contain the zero vector, be closed under addition and be closed under scalar multiplication

37
Q

Define column space and null space

A

Column space is the set of all linear combinations of the columns of A, the range of the transformation represented by A the span of the vectors that form the column of A these are all the column space. The null space is the set of all solutions to the equation Ax=0

38
Q

In the equation Ax=b what is the representative of the column space and the null space respectively

A

The column space is all bs whereas the null space is all xs.

39
Q

Describe the process for determining a basis for the column space of a matrix A.

A

Find the REF then identify pivot columns. Corresponding columns from the original matrix are the basis for the column space of a.

40
Q

What are the two properties that are basis must have?

A

The vectors in the basis must be literally independent and span the subspace

41
Q

Describe the process for determining a basis for the nail space of a matrix A.

A

Compute the solution set to Ax=0 form an augmented matrix, find RREF, write parametric form.

42
Q

How many ways can I get in vector be written with respect to a basis?

A

Only one. Consider Cartesian coordinates.

43
Q

The number of vectors in a basis is referred to as what?

A

The dimension

44
Q

What is the term for the dimension of the column space of matrix A

A

Rank

45
Q

What is the term for the dimension of the nail space of matrix A?

A

Nullity

46
Q

Solution set

A

Set of all possible solutions of a linear system

47
Q

Equivalent

A

Two linear systems are called equivalent if they have the same solution set

48
Q

Inconsistent

A

No solutions

49
Q

Three basic operations to simplify linear systems

A

1) replace one equation by the sum of itself and a multiple of another equation
2) interchange two equations
3) multiply all the terms in an equation by a non-0 constant

50
Q

Elementary row operations

A

One replace
Two interchange
Three scaling
Note: they are reversible

51
Q

Row Equivalent

A

Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. They will have the same solution set.

52
Q

Is the system consistent?

A

Does at least one solution exist?

53
Q

Is the solution unique?

A

If a solution exists, is it the only one?

54
Q

Leading Entry

A

The left most non-0 entry in a non-0 row

55
Q

Non-zero row or column

A

Any column/row in a matrix that contains at least one and non-0 entry

56
Q

Three properties of REF

A

1) all non-0 rows are above any rows of all zeros.
2) each lead entry of a row is in a column to the right of the leading entry of the row above it
3) all entries in column below leading entry are zero.

57
Q

Properties of RREF

A

1) the leading entry in each row is one
2) each leading one is the only non-0 entry in its column.

58
Q

Is REf unique

A

No

59
Q

Is RREF unique

A

Yes

60
Q

Row Echelon form of A

A

If a matrix a is row equivalent to an echelon matrix you, then U is a row, equivalent form of A

61
Q

The row reduced echelon form of A

A

If you is in a row, reduced echelon form the reduced echelon form of A

62
Q

Pivot position

A

The location in matrix that corresponds to a leading one in the RREF form

63
Q

Pivot column

A

A column of a matrix that contains a pivot position

64
Q

Basic variables

A

Variables which correspond to pivot columns in the matrix

65
Q

Free variables

A

Variables from non-pivot columns

66
Q

In a parametric description, what are free variables?

A

Free variables, active parameters

67
Q

Column vector, vector

A

Matrix with only one column

68
Q

R2

A

The set of all vectors with two entries

69
Q

Zero vector

A

Vector whose entries are all zero, denoted by 0

70
Q

Asking if B is in the span of {v1…vp}

A

If x1v1+…+xpvp = b

71
Q

Homogeneous

A

Ax=0

72
Q

Ax= 0 facts

A
  • Always has the trivial solution
  • Has non trivial solution if more than 1 free variable.
73
Q

Domain

A

Rn

74
Q

Codomain

A

Rm

75
Q

One to one

A

If each b in Rm is the image of at most one x in Rn. Is a uniqueness question. Needs pivot position in every row

76
Q

Determinant if 2x2 matrix

A

ad-bc, can’t equal 0

77
Q

Three properties of subspaces in Rn

A

1) Zero vector is in subspace
2) For each u and v, the sum of u+v is in subspace
3) For each u in H and each scalar c, the vector cu is in H

Linear Algebra (2 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions