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Linear Algebra > Linear Algebra > Flashcards

Linear Algebra Flashcards

1
Q

Define hyper-plane

A

More than 3 variables

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2
Q

Define shew lines

A

Lines in a 3 dimensional space that don’t intersect and are not parallel

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3
Q

Define homogenous system

A

A system of equations that all equations equal 0

All constant matrix solutions are 0 in an augmented matrix

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4
Q

Show a system of linear equations

A

a11x1 +… + a1nxn = b1
am1x1 + … + amnxn = bm

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5
Q

Define scaler

A

Multiplying a real number with a variable

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6
Q

What is a solution set

A

All the set of possible solutions

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7
Q

If a system of equations has at least one solution, what is it called

A

Consistent

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8
Q

How do you solve with elemental operation

A

Add or subtract 2 or more equations to remove variables and then back substitution

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9
Q

What are the elementary operations

A

Interchange order of equations
Multiple an equation by a non-zero number
Replace any equation with itself after adding it to a multiple of another equation

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10
Q

Solve using elementary operations

X + 3y + 6z = 25
2x + 6y + 14z = 58
2y + 5z = 19

A

X = 1
Y = 2
Z = 3

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11
Q

What is an augmented matrix

A

A matrix with both a coefficient matrix and a constant matrix separated by a line

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12
Q

How are elementary row operations different from elementary operations

A

elementary row operations are in the form of a matrix

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13
Q

Define leading entry

A

First non-zero entry of a row when going left to right

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14
Q

Rules followed to create row-echelon form

A

All non-zero rows are above any rows of zeros

Leading entry of a row is in a column to the right of the leading entry before it, and they are equal to 1

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15
Q

Difference between row-echelon form and reduced row-echelon form

A

In reduced row-echelon form, all entries above and below the leading entries are zero

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16
Q

What is a pivot position

A

Location of a leading entry in a row-echelon form

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17
Q

Define pivot column

A

Column that contains a pivot position

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18
Q

Create row-echelon and reduced row-echelon form of:

0 -5 -4
1 4 3
5 10 7

A

1 4 3
0 -5 -4
0 0 0

1 0 -1/5
0 1 4/5
0 0 0

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19
Q

What if all variable matrix entries and the constant matrix of an equation equal 0

A

Infinite solutions

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20
Q

What if all variable matrix entries of an equation equal 0 but the constant matrix entry isn’t zero

A

No possible solution

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21
Q

What if there are more variables than equations

A

Rather unlimited or zero soutions

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22
Q

What are the variables of a matrix not in a pivot column

A

A parameter

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23
Q

Difference in solving Gaussian elimination and gauss-jordan elimination

A

Gaussian elimination uses back substitution to solve variables

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24
Q

What is a basic variable

A

A variable entry that is not a parameter

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25
Q

What is a free variable

A

A variable entry that is a parameter

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26
Q

What is an equivalent matrix

A

A matrix that can be created from another matrix using elementary row operations

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27
Q

Why do homogenous systems have a trivial solution

A

The variables can all be zero

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28
Q

Define basic solution

A

Columns created using the parameters of a solution set

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29
Q

Find basic solutions

X + 4y + 3z = 0
3x + 12 y + 9z = 0

A

x1 = [-4] x2 = [-3]
[1 ] [0 ]
[0 ] [1 ]

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30
Q

What is a linear combination

A

Adding column matrixes together

V = a1X1 + … + anXn

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31
Q

Define rank

A

Rank is the number of rows with leading entries in a row-echelon for.

32
Q

m x n Coefficient matrix of a homogenous system has how many parameters

A

n-r parameters

33
Q

With a homogenous system how is there only one solution

A

If the systems rank equals the coefficient columns

34
Q

Balance:
KOH + H3PO4 –> K3PO4 + H2O

A

3KOH + 1H3PO4 –> 1K3PO4 + 3H2O

35
Q

In a matrix, how does (m x n) compare to (i,j) in a matrix

A

m and i are rows
n and j are columns

36
Q

What are individual elements of a matrix called

A

Entries or components

37
Q

What is a zero matrix

A

Matrix with only zeros

38
Q

What matrixes can be added together

A

Matrixes of the same dimensions

39
Q

What is the additive inverse of A

A

-A

40
Q

What are the two types of vectors

A

Column and row vectors

41
Q

What is the vector form of a system of equations

A

Column vectors with the variables being “scalars”

42
Q

What is the matrix form of a system of equations

A

AX = B

All being matrixes

43
Q

What must be true to multiple 2 matrixes

A

the first # of matrix column and second # of matrix rows, must be equal

44
Q

What is the resulting (m x n) of multiplying 2 matrixes

A

the first # of matrix rows by the second # of matrix columns

45
Q

Show A x B =

A=[1 2 1] B=[1 2 0]
[0 2 1] [0 3 1]
[-2 1 1]

A

[-1 9 3]
[-2 7 3]

46
Q

When multiplying 2 matrixes. How can you find a specific (i,j) solution without solving the rest of the matrix.

A

multiple of the first row element of the i’th row of the first matrix with the first column element of the j’th column of the second matrix. Then continue this with every element in the i’th row of the first matrix with the the j’th column of the second matrix. Then add them all together.

47
Q

define commute

A

two matrixes equal the same multiplied forwards or backwards

48
Q

what is the transpose of a matrix

A

When you swap the rows and columns of a matrix or (i,j) –> (j,i)

49
Q

(A^T)^T =

A

A

50
Q

(AB)^T =

A

(B^T)(A^T)

51
Q

(rA + sB)^T =

A

rA^T + sB^T

52
Q

A(rB + sC) =

A

r(AB) + s(AC)

53
Q

(B + C)A =

A

BA + CA

54
Q

A(BC) =

A

(AB)C

55
Q

k(A+B) =

A

kA + kB

56
Q

k(pA) =

A

(kp)A

56
Q

(k+p)A =

A

kA + pA

57
Q

IA =

A

A

58
Q

A+B =

A

B+A

59
Q

(A+B) + C =

A

A + (B+C)

60
Q

A+0 =

A

A

61
Q

A+(-A) =

A

0

62
Q

When is a matrix symmetric

A

When A = A^T

63
Q

When is a matrix skew symmetric

A

When A = -A^T

64
Q

What is an identity matrix

A

a square matrix with 1’s down the main diagonal and zeros in all other elements

Kronecker symbol is δ δij = 1 if i=j , 0 if i<>j

65
Q

Why does the position of a identity matrix matter and what kind of identity does this prove an identity matrix is

A

AIn=A & ImA=A

multiplicative identity

66
Q

When is a matrix considered invertible

A

If A(A^-1) = (A^-1)A = I

67
Q

If AB = BA = I , what does this mean

A

B = A^-1

68
Q

Find the inverse of [1 2 2]
[1 0 2]
[3 1 -1]

A

[-1/7 2/7 2/7]
[1/2 -1/2 0]
[1/14 5/14 -1/7]

69
Q

What is the easiest way to solve AX=B if possible

A

X=(A^-1)B

70
Q

(A^T)^-1 =

A

(A^-1)^T

71
Q

(AB)^-1 =

A

(B^-1)(A^-1)

72
Q

(A1*A2…Ak)^-1 =

A

(Ak^-1)…(A2^-1)(A1^-1)

73
Q

(A^-1)^-1

A

A

74
Q

I^-1 =

A

I

75
Q

(A^k)^-1 =

A

(A^-1)^k

76
Q

(pA)^-1 =

A

(1/p)A^-1

Linear Algebra (1 decks)

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