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Linear Algebra > All linear algebra > Flashcards

All linear algebra Flashcards

1
Q

What is a linear equation?

A

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

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2
Q

What is a system of linear equations?

A

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

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3
Q

What is a solution of a system of linear equations?

A

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

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4
Q

What is the solution set of a linear system?

A

The set of all possible solutions

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5
Q

When are two linear systems equivalent?

A

When they have the same solution set

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6
Q

What types of solutions can a linear system have?

A

1 - No solution
2 - One solution
3 - Infinitely many solutions

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7
Q

When is a linear system consistent?

A

When it has either one solution or many solutions

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8
Q

When is a linear system inconsistent?

A

When it has no solutions

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9
Q

What is a matrix?

A

A rectangular array

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10
Q

What is a coefficient matrix?

A

A matrix containing the coefficients of a linear system

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11
Q

What is an augmented matrix?

A

The coefficient matrix with an added column containing constants from the right sides of a system of linear equations

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12
Q

What are the elementary row operations?

A

1- Replacement - Replace one row by the some of itself and a multiple of another row
2- Interchange - Interchange two rows
3- Scaling - Multiply all entries in a row by a nonzero constant

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13
Q

When are two matrices row equivalent?

A

If there is a sequence of elementary row operations that transforms one matrix into the other.

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14
Q

When do two systems have the same solution set?

A

When the augmented matrices of the two systems are row equivalent.

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15
Q

When is a rectangular matrix in echelon form?

A

1- All nonzero rows are above any rows of all zeroes
2- Each leading entry of a row is in a column to the right of the leading entry of the row above it
3- All entries in a column below a leading entry are zeroes

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16
Q

When is a rectangular matrix in reduced echelon form?

A

When it is in echelon form, AND:
1- The leading entry in each nonzero row is 1
2- Each leading 1 is the only nonzero entry in its column

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17
Q

What is Theorem 1?

A

Uniqueness of the Reduced Echelon Form:

Each matrix is row equivalent to one and only one reduced echelon matrix.

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18
Q

What is a pivot position in a matrix A?

A

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

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19
Q

What is a pivot column of matrix A?

A

A column of A that contains a pivot position

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20
Q

What is a pivot?

A

A nonzero number in a pivot position that is used as needed to create zeroes via row operations

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21
Q

What is the forward phase of the row reduction algorithm?

A

The steps required to bring it into echelon form

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22
Q

What is the backward phase of the row reduction algorithm?

A

The steps required to bring it from echelon form to reduced echelon form

23
Q

What are basic variables?

A

Variables that correspond to pivot columns in a matrix

24
Q

What are free variables?

A

Variables for which you are free to choose any value

25
Q

What is Theorem 2?

A

Existence and Uniqueness Theorem:
A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column (coefficient matrix has no row all 0s with the corresponding augmented column being nonzero).

If a linear system is consistent, then the solution set contains either (1) a unique solution, when there are no free variables (2) infinitely many solutions, when there is at least one free variable.

26
Q

What is a column vector (or vector)?

A

A matrix with only one column

27
Q

When are two vectors in R^2 equal?

A

When and only when their corresponding entries are equal

28
Q

What is the parallelogram rule for addition?

A

If u and v in R^2 are represented as points in the plane, then u+v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.

29
Q

What is the zero vector?

A

The vector whose entries are all zero

30
Q

What are the algebraic properties of of R^n?

A 
1- u + v = v + u
2- (u + v) + w = u + (v + w)
3- u + 0 = 0 + u = u
4- u + (-u) = -u + u = 0
5- c(u+v) = cu + cv
6- (c+d)u = cu + du
7- c(du) = (cd)u
8- 1u = u
31
Q

What is a linear combination?

A

Given vectors v1 v2 … vp in R^n and given scalars c1 c2 … cp, the vector y defined by y = c1v1 + … + cpvp

32
Q

What are the weights of a linear combination?

A

The constant coefficients on each vector

33
Q

How do you tell if a linear combination is valid?

A

A vector equation x1a1 + x2a2 + … + xnan = b has the same solution set as the linear system whose augmented matrix is [a1 a2 … an b]. b can be generated by a linear combination of a1 … an iff there exists a solution to the linear system corresponding to that augmented matrix.

34
Q

What is Span(v1 … vp)?

A

The collection of all vectors that can be written in the form c1v1 + c2v2 + … cpvp with c1 .. cp scalars.

35
Q

What is the set of all linear combinations of v1 … vp denoted by Span(v1 … vp) called?

A

The subset of R^n spanned (or generated) by v1 … vp

36
Q

What is the product of A and x Ax?

A

The linear combination of the columns of A using the corresponding entries in x as weights.

37
Q

What is an equation of the form Ax=b called?

A

A matrix equation

38
Q

What are the two points of Theorem 3?

A

If A is an mxn matrix with columns a1…an and if b is in R^m, the matrix equation Ax=b:

  1. Has the same solution set as the vector equation x1a1 + x2a2 + … + xnan = b
  2. Has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 … an b]
39
Q

When does the equation Ax=b have a solution?

A

Iff b is a linear combination of the columns of A

40
Q

What are the 4 logically equivalent points of Theorem 4?

A
  1. For each b in R^m, the equation Ax=b has a solution
  2. Each b in R^m is a linear combination of the columns of A
  3. The columns of A span R^m
  4. A has a pivot position in every row
41
Q

What is the row-vector rule for computing Ax?

A

If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.

42
Q

What are the two points of Theorem 5?

A

If A is an mxn matrix, u and v are vectors in R^n, and c is a scalar, then:

  1. A(u + v) = Au + Av
  2. A(cu) = c(Au)
43
Q

When is a system of linear equations homogeneous?

A

If it can be written in the form Ax=0.

44
Q

How many solutions does a system Ax=0 have?

A

At least one

45
Q

Under what circumstances does the homogeneous equation Ax=0 have a nontrivial solution?

A

If and only if the equation has at least one free variable

46
Q

How can the solution set of a homogeneous equation Ax=0 always be expressed, explicitly?

A

Span(v1 … vp)

47
Q

Under what circumstances is the solution set of Ax=0 a line through the origin?

A

If Ax=0 has only one free variable

48
Q

Under what circumstances is the solution set of Ax=0 a plane through the origin?

A

If Ax=0 has two or more free variables

49
Q

What equation describes the solution set of Ax=0 in parametric vector form?

A

x = tv

50
Q

What equation describes the solution set of Ax=b in parametric vector form?

A

x = p + tv

51
Q

What is the equation of the line through p parallel to v?

A

x = p + tv

52
Q

What is the solution set of Ax=b?

A

A geometric figure (line, plane) through p parellel to the solution set of Ax=0

53
Q

What is Theorem 6?

A

If Ax=b is consistent for some given b, and p is a solution, then the solution set of Ax=b is the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax=0

Linear Algebra (1 decks)

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