Midterm 1 Review

Question 1

Echelon form criteria

Answer 1

1. Zero rows at the bottom
2. Leading entries are down and to the right (Staircase)
3. Zeros below all leading entries (Doesnt have to be 0 on top tho)

Question 2

Consistent

Answer 2

No row of 0’s that equals a number

Question 3

Inconsistent

Answer 3

There is a row of 0’s that equals a number

Question 4

Arbitrary

Answer 4

the variable can be anything

Question 5

Linear combination

Answer 5

cnVn + CnVn (Combining all multiples of the vector)

Question 6

Span

Answer 6

Set of all liner combinations

Question 7

How to tell if one vector is in the span of other vectors

Answer 7

Set up an augmented matrix, put it in echelon form, and see if it’s consistent

Question 8

The rule when multiplying a matrix and a vector

Answer 8

The number of columns in A has to equal the number of rows in the vector

Question 9

How big will a vector be after multiplying a matrix and a vector?

Answer 9

the number of columns in A x 1

Question 10

How to tell if Ax=b has a solution

Answer 10

Set up an augmented matrix and check if it is consistent

Question 11

Vector Equation

Answer 11

x1[vector] + x2[vector] + x3[vector]=[vector]

Question 12

Matrix equation

Answer 12

[matrix]x[column vector] = [column vector]

Question 13

Homogenous

Answer 13

Can be written in the form Ax =0

Question 14

Trivial solution

Answer 14

zero vector x=0

Question 15

Nontrivial solution

Answer 15

a nonzero vector x that satisfies the Ax=0

Question 16

How to tell if Ax=0 has a non-trivial solution

Answer 16

If it has a free variable

Question 17

How to tell if Ax=0 has a trivial solution

Answer 17

It has no free variables

Question 18

Nonhomogenous system

Answer 18

in Ax = b, b does not equal 0

Question 19

Parametric vector form

Answer 19

[collumn vector]x1 +[collumn vector]x2 + [collumn vector]x3

Question 20

How to find the nullity of a matrix

Answer 20

How many free variables does ti have

Question 21

0 dimensions

Answer 21

Point

Question 22

1 dimension

Answer 22

Line

Question 23

2 dimensions

Answer 23

Plane

Question 24

How to tell if a set of vectors is linearly independent

Answer 24

The trivial solution is the only solution to the vector equation or it has no free variables

Question 25

How to tell if a set of vectors is linearly dependent?

Answer 25

has at least one non trivial solution when = to 0 or there are free variables

Question 26

Other methods of detecting linear dependence

Answer 26

1. a 0 vector
2. multiple or sum of other vectors
3. more vectors than there are elements =
4. There is a free variable

Question 27

3 pivot positions

Answer 27

entire span of r^3

Question 28

2 pivot positions

Answer 28

plane in r^3

Question 29

1 pivot position

Answer 29

line in R^3