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Linear Algebra > Lecture Chapter 1 > Flashcards

Lecture Chapter 1 Flashcards

1
Q

What is a solution set?

A

The set of all possible solutions of a linear system

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2
Q

What are equivalent systems?

A

They have the same solution set

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3
Q

What are the possibilities of solutions of a system

A

either 1, zero or infinitely many solutions

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4
Q

What does consistent mean? inconsistent?

A

consistent : one or infinitely many solutions

inconsistent : no solution

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5
Q

How many rows and columns has a mxn matrix?

A

m rows, n columns

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6
Q

three characteristics of echelon form

A
  • all nonzero rows are above any rows of all zeros
  • each leading entry is on the right of the one above
  • all entries below a leading entry are zeros
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7
Q

three characteristics of reduced echelon form

A
  • echelon form conditions
  • the leading entry in each nonzero row is 1
  • each leading 1 is the only nonzero in its column
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8
Q

What is a pivot position

A

leading 1 in the reduced echelon form

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9
Q

What are row equivalent matrices

A

if row operations can transform one into the other

the two systems are equivalent (same solution set)

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10
Q

Can a matrix be row equivalent to multiple reduced echelon matrices

A

no, only one

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11
Q

what characterizes a consistent linear system

A

no [0 0 0 … 0 / b] in the echelon form of the augmented matrix

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12
Q

What is a linear combination

A

vector y = c1v1 + c2v2 + … + cpvp

linear combination of v1,v2,…,vp with weights c1,c2,…,cp

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13
Q

How do you find the solution of a vector equation x1a1 + x2a2 + … + xnan = b

A

the equation has the same solution set as the linear system with augmented matrix [a1 a2 … an / b]

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14
Q

What is Span {v1 v2 … vp} ?

A

The subset of Rn spanned by v1, v2, … , vp

The set of all linear combinations of v1, v2, … , vp (collection of vectors)

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15
Q

What is Ax

A

Linear combination of the columns of A using the corresponding entries in x as weights

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16
Q

How do you find the solution of Ax=b?

A

same solution set as
x1a1 + x2a2 + … + xnan = b
[a1 a2 … an b]

17
Q

What statements are equivalent to ‘for each b in Rm, Ax=b has a solution’

A
  • each b in Rm is a linear combination of the columns of A
  • the columns of A span Rm
  • A has a pivot position in every row
18
Q 
A(u+v) = Au + Av       
A(cu) = c(Au)
A
19
Q

What is a homogeneous linear system?

A

Ax=0

20
Q

Two properties of Ax=0 (homogeneous linear system)

A
  • always has at least one solution : trivial solution : zero vector 0
  • if Ax=0 has a nontrivial solution = equation has at least one free variable
21
Q

parametric vector equation of a line through the origin

A

x = su

22
Q

parametric vector equation of a plane through the origin

A

x = su + tv

23
Q

parametric vector equation of a line not through the origin

A

x = su + p

24
Q

parametric vector equation of a plane not through the origin

A

x = su + tv + p

25
Q

When is a set of vectors linearly independent?

A

When the vector equation x1v1 + x2v2 + … + xnvn = 0

has only the trivial solution

26
Q

When is a set of vectors linearly dependent? What is the linear dependence relation?

A

When there exists weights not all zero such that
c1v1 + c2v2 + … + cpvp = 0
(linear dependence relation)

27
Q

When are the columns of A linearly independent?

A

If Ax=0 has only the trivial solution

28
Q

What can you say about a set containing one vector linearly dependent?

A

It’s the zero vector

29
Q

What can you say about a set containing two vectors linearly dependent?

A

The vectors are a multiple of one another

30
Q

What can you say about a set containing two or more vectors linearly dependent?

A

At least one is a linear combination of the others

31
Q

What can you say about a set containing more vectors than there are entries in each vector

A

The set is linearly dependent

32
Q

What can you say about a set containing the zero vector

A

The set is linearly dependent

33
Q

What can you say about a transformation T from Rn to Rm

A
  • assigns to each x in Rn a T(x) in Rm
  • Rn domain
  • Rm codomain
  • T(x) in Rm is the image of x in Rn
  • The set of all images is the range of T
34
Q

What is T(x)=Ax

A

matrix transformation

35
Q

Three properties to show a transformation T is linear

A
  • T(u+v) = Tu + Tv
  • T(cu) = cT(u)
  • T(0) = 0
36
Q

What is an identity matrix

A

1 in the main diagonal, zeros elsewhere

37
Q

When is T : Rn–>Rm onto Rm?

A

if each b in Rm is the image of at least one x in Rn

38
Q

When is T : Rn–>Rm one-to-one?

A

if each b in Rm is the image of at most one x in Rn

  • -> T(x)=0 only has the trivial solution
  • -> the columns of A are linearly independent
39
Q

When does T maps Rn onto Rm?

A

When the columns of A span Rm

Linear Algebra (10 decks)

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