Midterm 1

Question 1

Equation of a line in a plane

Answer 1

ax + by = c

Question 2

Compatible

Answer 2

equation = # variables

1 solution
# equations = # variables

Question 3

Incompatible

Answer 3

equations = # variables

0 solution
[0 0 0 … 1]

Question 4

Compatible indeterminate

Answer 4

equations < # variables

infinite solutions
# equations < # variables

Question 5

Elementary row operations

Answer 5

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

Question 6

Vector

Answer 6

(m x 1) matrix

Question 7

RREF rules

Answer 7

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

Question 8

Is matrix addition associative?

Answer 8

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

Question 9

Is matrix addition commutative?

Answer 9

Yes
A+B=B+A

Question 10

What is the transpose of a matrix?

Answer 10

Interchange rows & columns

Question 11

What does it mean for a matrix to be symmetric?

Answer 11

Square matrix is equal to its transpose

Question 12

Span

Answer 12

set of vectors you can reach with a linear combinations

Question 13

(A^T)T =

Answer 13

(A^T)T = A

Question 14

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

Answer 14

1. Equation: x1v1 + … + xnvn = b
2. System: Ax = b
3. Augmented matrix: [a1 … an | b]

Question 15

Linearly dependent:

Answer 15

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

Question 16

Linearly independent:

Answer 16

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

Question 17

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

Answer 17

linearly dependent

Question 18

Is the zero vector linearly indpendent or dependent?

Answer 18

linearly dependent

Question 19

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

Answer 19

linearly independent

Question 20

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

Answer 20

Yes

Question 21

Rank

Answer 21

linearly independent rows of the matrix

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

Question 22

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

Answer 22

NO

Question 23

(AB)^T =

Answer 23

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

Question 24

Homogeneous linear system

Answer 24

Ax = 0

Solution set sends vectors to zero vector

Question 25

Non-homogeneous linear system

Answer 25

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

Question 26

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

Answer 26

ka

Question 27

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

Answer 27

a1 + a2

Question 28

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

Answer 28

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

Question 29

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

Answer 29

c+a solution to Ax=b

Question 30

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

Answer 30

c1 - c2 is a solution to Ax=0

Question 31

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

Answer 31

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

Question 32

Any matrix transformation can be written as what?

Answer 32

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

Question 33

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

Answer 33

T(0) = 0

Question 34

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

Answer 34

T(x) = Ax

Question 35

What is the matrix of rotation by angle theta?

Answer 35

Question 36

Invertible Matrix

Answer 36

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

Question 37

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

Answer 37

det(a) ≠ 0 otherwise space would be squashed into a lower dimension and there is no inverse transformation that could undo that

Question 38

How can you find the inverse of a matrix?

Answer 38

[ A | I ] --> RREF [I | A^1] Augment it with the identity and do row operations

Question 39

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

Answer 39

- 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

Question 40

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

Answer 40

Only have an inverse on one side Left inverse: iff it has linearly independent columns Right inverse: iff it has linearly independent rows

Question 41

Vector space

Answer 41

A set of vectors V which has: - u + v in V - ku in V Must satisfy 8 axioms

Question 42

What are the 8 axioms of a vector space?

Answer 42

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