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LINEAR ALGEBRA > TEST (Ch 2 - 3) > Flashcards

TEST (Ch 2 - 3) Flashcards

1
Q

Matrix is invertible iff det(A) ≠ ?

A

det(A) ≠ 0

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2
Q

Inverse of a (2x2) matrix:

A
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3
Q

The determinant can only be calculated for what kind of matrices?

A

SQUARE matrices (nxn)

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4
Q

What is the det of a (1x1) matrix?

A

det(A) = A

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5
Q

What is the determinant of a (2x2) matrix?

A

det(A) = ad - bc

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6
Q

What is the minor of entry aij (Mij)?

A

the det of submatrix that remains after the ith row and the jth column are deleted from A.

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7
Q

What is the cofactor of entry aij (Cij)?

A
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8
Q

What is the relationship between a minor and its corresponding cofactor?

A

They are either the same or negatives of each other

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9
Q

What is cofactor expansion?

A

obtained by multiplying the entries in any row/column by the cofactors and adding the resulting product = determinant

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10
Q

Does it matter which row/column you choose when doing cofactor expansion?

A

Get the same determinant no matter the row/column you choose

Easiest: choose the one with the most 0s

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11
Q

What is the determinant of a triangular matrix?

A

det(A) = product of entries on the main diagonal

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12
Q

What is a simple technique for evaluating the det of a (2x2) or (3x3) matrix?

A
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13
Q

If a matrix has a row or column of 0s, what is its det?

A

det(A) = 0

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14
Q

If a matrix has a 2 proportional rows/columns, what is its det?

A

det(A) = 0

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15
Q

What is the det(AT)?

A

det(AT) = det(A)

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16
Q

Elementary Row Operations on Determinants

If single row/column was multiplied by k, what is det?

A

det(B) = k det(A)

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17
Q

Elementary Row Operations on Determinants

If single two rows/columns were interchanged, what is det?

A

det(B) = –det(A)

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18
Q

Elementary Row Operations on Determinants

If a multiple of a row/column was added to another, what is det?

A

det(B) = det(A)

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19
Q

How can you evaluate the det using Row Reduction?

A
  1. Reduce matrix to REF (upper triangular)
  2. Cofactor expansion or main diagonal
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20
Q

Is there an addition formula for determinants?

A

NO!

det(A+B) ≠ det(A) + det(B)

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20
Q

What is the scalar multiplication formula for determinants?

A

det(kA) = kn det(A)

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21
Q

What does invertible mean?

A

it can be expressed as a product of elementary matrices

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22
Q

If A and B are square matrices of the same size:

det(AB) =

A

det(AB) = det(A) det(B)

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23
Q

What is the det(A-1)?

A
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24
Q

What is an adjoint?

A

Transpose of the matrix of cofactors of A

25
Q

What is Cramer’s Rule?

A
  • helps solve for UNIQUE solution of a system
  • First double-check that det(A) ≠ 0
26
Q

Vectors are equivalent if ___

A

IF they have the same magnitude and direction

Or if all of their respective components are equal

27
Q

What is a zero vector?

A

Initial & terminal points coincide has length 0

Has no natural direction, it can be assigned to any direction that’s convenient

28
Q

What is the parallelogram rule (+ and – vectors)?

29
Q

What is the difference between parallel & collinear with vectors?

A

Parallel = collinear for vectors → because parallel vectors can be moved to become collinear

30
Q

Component form of vectors

31
Q

What is the formula for a vector whose initial point is NOT the origin?

32
Q

N-space (Rn)

A

set of ordered n-tuples

33
Q

Algebraic properties of vectors:

A

prove using components

34
Q

What is a linear combination?

A

The ks are the coefficients of the linear combination

35
Q

What is a norm (|| ||) of a vector?

A

length/magnitude of vector

36
Q

Properties of norms

37
Q

What is a unit vector?

A

vector magnitude = 1

38
Q

Standard unit vector

A

v = (v1, v2, v3 … vn)

39
Q

What is the distance between two vectors?

40
Q

Dot product definition

A

Product of magnitude of component v in the direction of w with the magnitude of w.

41
Q

If two vectors are oriented very similarly then there dot product is __

A

The more similarly two vectors are oriented the higher the dot product

42
Q

Geometric definition of dot product

43
Q

What does the dot product tell us about the angle between two vectors?

44
Q

Algebraic definition of the dot product

45
Q

Expressing the norm of a vector with dot product

46
Q

Properties of the dot product

47
Q

Cauchy-Shwarz Inequality

48
Q
A

like a triangle (no side can be longer than the sum of the other sides)

49
Q

Two vectors are orthogonal if __

A

dot product = 0

50
Q

Are 0 vectors orthogonal to anything?

A

Zero vectors in Rn are perpendicular to every vector in Rn

51
Q

What is a normal?

A

vector orthogonal to line/plane

52
Q

Normal: Point-normal form

53
Q

Normal: Component form

A

Through point P = (x0, y0, z0), n = (a, b, c)

54
Q

Vector form of a line/plane through the origin

55
Q

ax + by + c = 0 represents what?

A

a line in R2 with normal n = (a, b)

56
Q

ax + by + cz + d = 0 represents what?

A

a plane in R3 with normal n = (a, b, c)

57
Q

What is an orthogonal projection?

58
Q

Vector component of u along a

59
Q

Vector component of u orthogonal a

60
Q

Norm (|| ||) of a vector projection

LINEAR ALGEBRA (7 decks)

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