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Final Actually Flashcards

1
Q

Linear Equation definition

A

An equation that can be written in the form a1x1+ a2x2+…+anxn =b

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2
Q

Coefficients

A

Real of complex numbers that are multiplied by x in the linear equation

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3
Q

A system of linear equations (linear system)

A

A collection of one or more linear equations involving the same variables.

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4
Q

The solution of a system

A

Is a list of all numbers that makes each equation a true statement when the values of s1,s2,…sn are substituted for x1, x2,… xn respectively.

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5
Q

Solution set

A

Set of all possible solutions

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6
Q

Equivalent

A

If they have the same solution set

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7
Q

How many solutions can a system of linear equations have

A

1) no solutions
2) one solution
3) infinite solutions

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8
Q

Consistent

A

Has one solution or infinitely many solutions

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9
Q

Inconsistent

A

No solutions

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10
Q

Coefficient matrix

A

A matrix of the coefficient of a set of linear equations, does not include answers

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11
Q

Augmented Matrix

A

The coefficient matrix with an added column containing the constants from the right sides of the equations.

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12
Q

What are the dimensions of an mxn matrix

A

The matrix has m rows and n columns

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13
Q

Three basic operations to solve linear systems

A

1) Replace one equation by the sum of itself and a multiple of another equation
2) Interchange two equtions
3) Multiply all terms in an equation by a nonzero constant

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14
Q

Row equivalent

A

Two matrices are 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

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15
Q

Two fundamental questions of linear systems:

A

1) Is the system consistent; that is, does at least one solution exist?
2) If a solution exists, is it the only one; that is, is the solution unique?

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16
Q

Nonzero row or column

A

A row or columns that contains at least one nonzero entry

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17
Q

Leading entry

A

Leftmost nonzero entry in a nonzero row

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18
Q

Row Echelon Form REF properties

A
  1. All nonzero rows are above any rows of all zeros
  2. Each leading entry of a row is in a column to the tight of the leading entry of the row above it
  3. All entries in a column below a leading entry are zeros
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19
Q

Reduce Row Echelon form properties

A

Same properties as REF plus:
4)The leading entry in each nonzero row is 1
5) Each leading 1 is the only nonzero entry in its column
(Is unique)

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20
Q

Pivot position

A

A location in A that corresponds to a leading 1 in the reduced echelon form RREF of A

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21
Q

Pivot column

A

Column that contains a pivot position

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22
Q

Basic variables

A

Variables of x corresponding to pivot columns.

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23
Q

Free variables

A

Variables not corresponding to a pivot column

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24
Q

Parameters

A

Free variables in a parametric equation

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25
Column vector (vector)
A matrix with only one column
26
R^n
Denotes how many entries are in a column vector.
27
Scalar
A integer that is used to multiply a vector
28
Linear combination
A linear equation combined of scalars multplied by vectors y = c1v1 + c2v2+…+cpvp
29
How are span and consistency related.
“is b in the span {…}?” Is the same as asking “is Ax=b consistent?”
30
Homogeneous
A system is homogeneous if it can be written in the form Ax=0
31
Trivial solution
The solution x = 0, which always exists for Ax=0
32
Linearly Independent
If the vector equation has only the trivial solution.
33
Transformation (or function or mapping)
A transformation T from R^n to R^m is a rule that assigns to each vector x in R^n a vector T(c) in R^m.
34
Domain
The set R^n is the domain of T
35
Codomain
R^m is the codomain of T
36
Range
The set of all images T(x) is the range of T
37
Superposition principle
T(c1v1+…+cpvp) = c1T(v1)+…+cpT(vp)
38
Contraction
Given a scalar r, define T: R^2 -> R^2 by T(x) =rx. When 0<=r<=1.
39
Dilation
Given a scalar r, define T: R^2 -> R^2 by T(x) =rx. When r> 1
40
Onto
1) A mapping T: R^n -> R^m is said to be onto R^m if eacb b in R^m is the image if at least one x in R^n Or 2) T is onto R^m when the range of T is all the codomain of R^m
41
One to One
A mapping T: R^n -> R^m is said to be one to one if each b in R^m is the image of at most one x in R^n Has either a unique solution or none at all
42
Diagonal Matrix
Is a square nxn matrix whose non diagonal entries are 0.
43
Invertible
An nxn matrix A is said to be invertible if there is an nxn matrix c such that CA=I and AC=I.
44
Singular Matrix
A matrix that is not invertible
45
Nonsingular matrix
An invertible matrix
46
Determinant of a 2x2 matrix
det(A) = ad-bc
47
Subspace
A subspace of R^n is any set H in R^n that has three properties: 1) The zero vector is in H 2) For each unit and v in H, the sum u+v is in H. 3) For each u in H and each scalar c, the vector cu is in H.
48
Zero subspace
Only contains the zero vector
49
Column Space
The column space of a matrix A is the set Col A of all linear combinations of the columns of A.
50
Null space
The null space of a matrix A is the set Nul A of all solutions of the homogeneous equation Ax=0
51
Basis
A basis for a subspace H of R^n is a linearly independent set in H that spans H
52
Dimension if a subspace
The dimension of a nonzero subspace H, denoted by dim H is the number of vectors in any basis for H. The dimension of the zero subspace {0} is defined to be zero.
53
Rank of a matrix
The rank of a matrix A, denoted by rank A, is the dimension of the column space of A. (Or the number of pivot columns in A)
54
Rank of a matrix
The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.
55
Determinant of a 2 by 2 matrix
Det A = ad-bc
56
Minor of entry a_ij
Denoted by M_ij and is defined to be the determinant of the sub matrix that remains after the ith row and the jth column are deleted from A
57
Cofactor entry of a_ij
The number (-1)^(i+j) M_ij is denoted by C_ij and is called the cofactor of entry a_ij
58
Eigenvector
An eigenvector of an nxn matrix A is a nonzero vector x such that Ax = lambda(x) for some scalar lambda.
59
Eigenvalue
A scalar lambda is called an eigenvalue of A if there is a nontrivial solution x of Ax=lambda(x) such that an x is called an eigenvector corresponding to lambda
60
Characteristic eqution
The equation det(A-lambda*I)=0
61
Diagonalizable
A square matrix is diagonalizable if A is similar to a diagonal matrix that is if A=PDP^-1

Linear Algebra (5 decks)

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