New Other

Question 1

Unique Solution
Many Solutions
No Solution

Answer 1

No Free Variable
Free Variable
0 = #

Question 2

General Solution

Answer 2

{ x1 = 3x2 + 5
{ x2 is free

Question 3

Vector Equation

Answer 3

x1[3] + x2[8] = [1]
[5] + [3] = [2]

Question 4

Vector in span {v1, v2}

Answer 4

v = c1v1 + c2v2

Question 5

A times x

Answer 5

x1*A1 + x2*A2 + x3*A3
A1 = column 1

Question 6

Matrix Equation

Answer 6

[A] [x] = [b]

Question 7

Linearly Independent

Answer 7

rows <= columns
only trivial solution
no multiplies
no 0 vector

Question 8

T(x) is linear if…

Answer 8

T(0) = 0

Question 9

Standard Matrix

Answer 9

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

Question 10

One to One

Answer 10

One solution
Linearly Independent

Question 11

Inverse of 2x2

Answer 11

A^-1 = 1/detA [d -b, -c a]

Question 12

Solve x with Inverse

Answer 12

x = (A^-1) B

Question 13

Find Inverse

Answer 13

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

Question 14

Onto
Is Anxn inversable
Do the columns of A span R^n

Answer 14

Needs a pivot in every row

Question 15

Is triangle inversable

Answer 15

If diagonal multiple is non zero

Question 16

Subspace

Answer 16

Any set with a zero vector
H = span{ v1, v2, v3 }
u+v and cu in H

Question 17

Column Space

Answer 17

ColA = span{ a1, a2, a3 }
Rm, m = rows

Question 18

Null Space

Answer 18

All solutions for Ax=0
Rn, n = columns

Question 19

Basis for ColA

Answer 19

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

Question 20

Basis for NullA

Answer 20

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

Question 21

Basis for Subspace

Answer 21

The linearly independent set,
no matching

Question 22

Rank

Answer 22

RankA + dim(NullA) = # of columns

Question 23

Cofactor Expansion

Answer 23

detA = a11C11 + a12C12 + a13C13
Cij = (-1)^i+j detAij

Question 24

Row Operations

Answer 24

Addition, detB = detA, detE = 1
Multiplication, detB = k detA, detE = k
Swapping, detB = -detA, detE = -1

Question 25

Cramer’s Rule

Answer 25

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

Question 26

Adjugate

Answer 26

adjA =
C11 … Cn1
C1m … Cnm

Question 27

Adjugate Invertable Matrix Rule

Answer 27

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

Question 28

Cofactor

Answer 28

Cij = (-1)^i+j * detAij

Question 29

Coordinate Vector of x relative to B

Answer 29

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

Question 30

Parametric Form

Answer 30

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