Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Tufts - Linear > Midterm 1 > Flashcards

Midterm 1 Flashcards

1
Q

Equation of a line in a plane

A

ax + by = c

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Compatible

A

equation = # variables

1 solution
# equations = # variables

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Incompatible

A

equations = # variables

0 solution
[0 0 0 … 1]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Compatible indeterminate

A

equations < # variables

infinite solutions
# equations < # variables

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Elementary row operations

A

don’t change solution set
1. Interchanging two rows
2. Multiplying a row by a non-zero constant
3. Adding to a row a multiple of another

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Vector

A

(m x 1) matrix

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

RREF rules

A
  1. Leading 1s are the first # unless all 0 row
  2. Row with only 0s at the bottom
  3. Lower leading one must be farther to the right
  4. Each leading 1 column has 0s everywhere else in the column
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Is matrix addition associative?

A

Yes
(A+B)+C=A+(B+C)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Is matrix addition commutative?

A

Yes
A+B=B+A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

What is the transpose of a matrix?

A

Interchange rows & columns

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

What does it mean for a matrix to be symmetric?

A

Square matrix is equal to its transpose

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Span

A

set of vectors you can reach with a linear combinations

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

(A^T)T =

A

(A^T)T = A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What are three ways to solve a linear system of equations?

A
  1. Equation: x1v1 + … + xnvn = b
  2. System: Ax = b
  3. Augmented matrix: [a1 … an | b]
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Linearly dependent:

A

if there exists scalars (not all equal to zero) such that:
a1v1 + a2v2 + … + anvn = 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Linearly independent:

A

The only scalars such that a1v1 + a2v2 + … + anvn = 0 are all equal to zero

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

If a set of equations has more variables than equations, then is it linearly independent or dependent?

A

linearly dependent

18
Q

Is the zero vector linearly indpendent or dependent?

A

linearly dependent

19
Q

Given a matrix in REF, the non-zero rows are

A

linearly independent

20
Q

If two vectors are linearly independent are you perform elementary row operations on them, are they still independent?

A

Yes

21
Q

Rank

A

linearly independent rows of the matrix

leading 1s in the general solution of Ax=0 (REF)

22
Q

If A and B are matrices,
Does AB = BA?

A

NO

23
Q

(AB)^T =

A

(AB)^T = B^T A^T

24
Q

Homogeneous linear system

A

Ax = 0

Solution set sends vectors to zero vector

25
Q

Non-homogeneous linear system

A

Ax = b

Solutions to Ax=b are translations of the solutions to Ax=0

26
Q

If a is a solution to Ax = 0 and k is scalar, what is another solution to Ax = 0?

A

ka

27
Q

If a1 and a2 are solutions to Ax=0, then what is another solution to Ax=0?

A

a1 + a2

28
Q

More generally iff a1 and a2 are solutions to Ax=0 and k are scalars, then what?

A

Any linear combination (k1a1+…knan) is also a solution to Ax = 0

29
Q

If a is a solution to Ax=0 and c is a solution to Ax=b, then what is another solution to Ax=b?

A

c+a solution to Ax=b

30
Q

If c1 and c2 are solutions to Ax=b, then what is a solution to Ax=0?

A

c1 - c2 is a solution to Ax=0

31
Q

What is a linear transformation? T: Rn -> Rm such that

A
  1. T(v + w) = T(v) + T(w)
  2. T(kv) = kT(v)
32
Q

Any matrix transformation can be written as what?

A

A unique matrix times a vector
T(x) = Ax

33
Q

If T is a linear transformation then, T(0) =

A

T(0) = 0

34
Q

What is another way of writing the linear transformation T(x)?

A

T(x) = Ax

35
Q

What is the matrix of rotation by angle theta?

A
36
Q

Invertible Matrix

A

Square matrix such that:
AA^-1 = I and A^-1A = I

37
Q

What is the condition that tells us if a square matrix is invertible?

A

det(a) ≠ 0

otherwise space would be squashed into a lower dimension and there is no inverse transformation that could undo that

38
Q

How can you find the inverse of a matrix?

A

[ A | I ] –> RREF [I | A^1]

Augment it with the identity and do row operations

39
Q

For a square matrix, what statements are equivalent regarding inverses?

A
  • There exists a matrix B such that BA = I
  • There exists a matrix C such that AC = I
  • The matrix A has rank n (full)
  • For every vector b, the equation Ax=b has a unique solution
40
Q

Can you find the inverse of an (M x N) matrix

A

Only have an inverse on one side

Left inverse: iff it has linearly independent columns
Right inverse: iff it has linearly independent rows

41
Q

Vector space

A

A set of vectors V which has:
- u + v in V
- ku in V

Must satisfy 8 axioms

42
Q

What are the 8 axioms of a vector space?

A
  1. u + v = v + u
  2. u + (v + w) = (u + v) + w
  3. u + 0 = u
  4. u + (-u) = 0
  5. (k1k2)u = k1(k2u)
  6. (k1 + k2)u = k1u + k2u
  7. k(u + v) = ku + kv
  8. 1u = u

Tufts - Linear (3 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions