Term Test 1

Question 1

Systems of linear equations can have what 3 types of solutions?

Answer 1

1. No solution
2. One solution
3. Infinitely many solutions

Question 2

What is an inconsistent vs consistent matrix?

Answer 2

Inconsistent: No solution (row of zeroes = #)
Consistent: one or infinitely many solutions

Question 3

What is row echelon form?

Answer 3

Rows of zeroes are at the bottom

The leading entry of each row is to the right of the row above’s leading entry

Question 4

What is Reduced Row Echelon form?

Answer 4

REF
All leading entries are 1
Each leading entry is the only non zero in it’s column

Question 5

What are basic and free variables?

Answer 5

Basic variables are pivots

Free variables are non pivots and can be anything

Question 6

What is a linear combination?

Answer 6

It’s the sum of the products of constants and vectors (y=c1v1+c2v2+…)

Question 7

What is span?

Answer 7

The collection of all possible linear combinations of vectors AKA the set of vectors generated by v1,v2…

Question 8

What is the matrix equation?

Answer 8

Ax=b

Question 9

When can you multiply two matrices?

Answer 9

When the first matrix has the same number of columns as the second matrices rows

Question 10

What is a homogenous system?

Answer 10

When ax=o (it can never be inconsistent)

Question 11

What is parametric vector form?

Answer 11

When the answer is written as a sum of constants multiplied by column vectors

Question 12

What is a non homogenous system?

Answer 12

When ax=b (but still related to homogenous)

Question 13

What is the trivial and non trivial solution to homogenous systems?

Answer 13

Trivial: x=o

Non trivial: anything else that is a solution

Question 14

What is linear independence?

Answer 14

If the vector equation has only the trivial solution

Question 15

What is linear dependence?

Answer 15

If the vector equation has a non trivial solution (free variable makes dependent)

Question 16

In a transformation, which is the domain and codomain?

Answer 16

T: R^n -> R^m
R^n is the domain 
R^m is the codomain 
"T maps R^n to R^m"
"T(x) is the image of x under T"

Question 17

What is the range?

Answer 17

The set of all vectors that are outputs of a mapping T: R^n -> R^m

Question 18

What are linear transformations?

Answer 18

The multiplication by a matrix
AKA Matrix transformations
“x–>Ax”
“x maps to Ax”

Question 19

What are the 2 conditions of a linear transformation?

Answer 19

1. T(u+v)=T(u)+T(v)

2. T(cu)=cT(u)

Question 20

What is injective mapping?

Answer 20

A mapping T: R^n -> R^m is one-one if each b in R^m is the image of at most one x in R^n (no doubles on the dots in the right circle)

If T is not injective, T(x)=0 has more than the trivial solution, so its linearly dependent

Question 21

What is surjective mapping?

Answer 21

A mapping T: R^n -> R^m Is ONTO if each b in R^m is the image of at least one x in R^n (each dot in the right circle is used)