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Question 1

Linear Combinations

Answer 1

a1x1 + a2x2 + a3x3
Set of vectors where each vector is multiplied by a weight

Ax is a linear combination of the vectors in A

a1x1 + a2x2 + a3x3, a is a vector, x is a weight

Question 2

Span {v1, v2, v3}

Answer 2

“Set of all linear combinations”
v1 is a base but
2v1, 5v1, and -v1 are
also answers

The linear combinations are simply the solutions

Question 3

Linear Independence

Answer 3

Only has a trivial solution
v1 != 0, no 0 vector
v1 and v2 cant be multiples
n <= m

Question 4

Parametric Form

Answer 4

x = p + x3v
x = [#] + x3[# in x3]
_
x = 3 + x2[5]
x1 = 3 + 5x2

Question 5

Homogeneous

Answer 5

Ax = 0

Question 6

Identity Matrix

Answer 6

Square matrix
1’s on diaginal
0’s elsewhere (sans b)

Question 7

Consistancy

Answer 7

Not consistant if 0x = #

Question 8

m x n

Answer 8

m = rows
n = columns

Question 9

RREF

Answer 9

1’s mostly on diaginal
1’s are leading entries
1’s are the only non zero entry in column

Question 10

Pivot Posistion

Answer 10

Leading non zero number in row

Question 11

Row Operations

Answer 11

1. R1 == R2
2. R1 + 3R2 => R1
3. R1 * (1/2) => R1

Question 12

Uniqueness

Answer 12

Is the found solution the only one

Question 13

Domain

Answer 13

Entirety/span of R^n

Question 14

Co Domain

Answer 14

Entirety of R^m

Question 15

Trivial Solution

Answer 15

x = 0

Question 16

Solution Set

Answer 16

An x(s) in Ax=b

Question 17

Image

Answer 17

x after some transformation
T(x), Ax, b

Question 18

Range

Answer 18

span of R^m
Set of all images of T(x)

Question 19

Standard Matrix

Answer 19

A = [ T(e1), T(e2), T(e3) ]

Question 20

T: R^n -> R^m is onto R^m

Answer 20

R^n = R^m

Question 21

T: R^n -> R^m is one to one

Answer 21

If T(x) = 0 has one solution
Aka if A is linearly independent

Question 22

Diagonal Matrix

Answer 22

Non zeros only on main diagonal
Square matrix

Question 23

AB

Answer 23

A[ b1, b2, b3 ]
Amxn * Bnxp

Question 24

Transpose

Answer 24

Amxn becomes Anxm

Question 25

Inverse Matrix

Answer 25

A * A^-1 = ID

Question 26

2x2 Invertable Matrix Rule

Answer 26

detA = ad - bc != 0 then invert invertA = 1/detA [d -b, -c a]

Question 27

Elementary Matrix

Answer 27

An ID matrix after only one row operation

Question 28

Inverse Elementary Matrix

Answer 28

The matrix that turns E into a ID matrix

Question 29

(3x3+) Invertable Matrix Rule

Answer 29

[ A | I ] => [ I | A^-1]

Question 30

Subspace

Answer 30

Any set with a zero vector H = span{ v1, v2, v3 }

Question 31

Column Space

Answer 31

ColA = span{ a1, a2, a3 } R dimension and column height need to match

Question 32

Null Space

Answer 32

All solutions for Ax=0

Question 33

Basis for Subspace

Answer 33

The linearly independent set, no span

Question 34

Standard Basis

Answer 34

{ e1, e2, e3 }

Question 35

Basis for NullA

Answer 35

1. Ax = 0 2. RREF 3. Parametic form 4. Vectors go into { v1, v2, v3 }

Question 36

Basis for ColA

Answer 36

1. RREF 2. Take pivot columns, discard free columns 3. Use original A, not RREF

Question 37

Coordinate Vector of x relative to B

Answer 37

xB = [c1, c2, c3] x = c1v1 + c2v2 + c3v3 Find xB

Question 38

Dimensions of Subspace

Answer 38

dimH = Number of vectors in Basis

Question 39

Rank

Answer 39

Number of pivot columns (Dimensions of ColA)

Question 40

Rank Theroem

Answer 40

rankA + dim(NullA) = n n = # of A columns

Question 41

Determinats

Answer 41

detA = (-1)^i+j * aij * detAij detA = ad - bc

Question 42

Cofactor Expansion

Answer 42

detA = a11C11 + a12C12 + a13C13

Question 43

Triangular detA

Answer 43

Multiply the main diagional

Question 44

detA Row Operations

Answer 44

Addition (detB = detA) Swapping (detB = -detA) Multiplication (detB = k detB)

Question 45

Cramer's Rule

Answer 45

xi = detAi (b) / detA detA but replace column Ai with b

Question 46

Adjugate Invertable Matrix Rule

Answer 46

For square invert A^-1 = (1 / detA) adjA

Question 47

Adjugate

Answer 47

adjA = [aij]^T = C11 ... Cn1 C1m ... Cnm

Question 48

Area or Volumn of Matrix

Answer 48

detA

Question 49

S vector space

Answer 49

(-∞, ..., v-2, v-1, 0, v1, v2, ..., ∞) Written as a row

Question 50

P vector space

Answer 50

p(t) = a0 + a1 \* t + a2 \* t^2 + ... an \* t^n