Exam 1

Question 1

Linear Equation

Answer 1

an equation that can be written as:

ax1+bx2+…..nxn

where a,b..n are real or complex numbers

Question 2

Linear System

Answer 2

One of more linear equations involving the same variables.

Must have
1. No solution (Consistent)
2. 1 solution (Consistent)
3. Infinite Solutions (Inconsistent)

Question 3

matrix (elementary row operations)

Answer 3

1. Interchange
2. Scaling
3. Replacement

Question 4

Row Equivalence

Answer 4

Two matrices are row equivalent if there is a series of elementary row operations that transforms one matrix into another

This means that the matrices associated linear system has the same solution set

Question 5

Echelon Form

Answer 5

1. Non Zero Rows are above rows of all 0s
2. Traingle Shape – Leading rows go down left to right
3. Below leading entry are zeroes

Question 6

RRef : Row Reduced Echelon Form

Answer 6

What:
1. Leading Entry is 1
2. Leading entry is only non zero entry

Note: RReF is UNIQUE

Use:
1. Determining qualities of variables, determining solution sets

Question 7

Pivot Position

Answer 7

What: Corresponds to the position of a leading 1. A pivot column contains a pivot solution.

Use:
Basic Variables (have a solution essentially) correspond to pivot columns

Free variables do not correspond to pivot columns and can be any value

Question 8

A linear system is consistent if…

Answer 8

it’s corresponding matrix has no row of the form 000 b where b does not equal 0

Consistent system has a solution (1 or infinite)

Why relevant: The existence of a free variable does NOT mean infinite solutions + best way to determine no solution

Question 9

Vector

Answer 9

1. A matrix with one column
2. Two vectors are equal if and only if thier entries are the same
3. zero vector has zeroes in all positions

Question 10

Vector Operations

Answer 10

1. Scalar Multiplication
2. Vector Addition
3. Parellelogram Rule

Question 11

Linear Combo

Answer 11

Combination of a set of vectors with a set of scalars as weights

relevant: for matrices..

Question 12

span

Answer 12

what: the set of all linear combinations of v1, v2,v3
why: where can we go using these vectors?

Question 13

matrix equations

Answer 13

A x> = b> only has a solution if b is a linear combination of the columns of A (think of how multiplication works)

Question 14

Ax = 0

Answer 14

-homogenous if it can be written in that form
- always at least one solution (trivial solution) x = 0
- non trivial solution (one free variable)

Question 15

The consistent equation with Ax = b, let p be a solution

Answer 15

the solution set is w = p + vh where vh is any solution of the homogenous equation

repercussions: if there is only a trivial solution, then Ax = b is the unique solution

Question 16

Linear Independence

Answer 16

A set of vectors is linearly indepedent if none of the vectors in the set are linear combinations of the others

1. Linearly independent vectors only have the trivial solution
2. Does not contain the zero vectors
3. More vectors than entries in vectors (more columns than rows)

Question 17

Given an mxn matrix, for each b in Rm Ax = b has a solution

Answer 17

1. b must be a linear combo of A
2. columns of A span Rm
3. pivot position in each row

Question 18

Codomain

Answer 18

all possible outputs

Question 19

range

Answer 19

all outputs

Question 20

Onto

Answer 20

Range and Codomain are the same
rn –> rm
Every b in rm is the image of at least one x in rn

Question 21

one to one

Answer 21

every b in Rm is the image of at MOST on x in Rn

Question 22

Linear Transformation requirement

Answer 22

1. vector addition
2. scalar multiplication

Question 23

Linear Transformation Properties

Answer 23

1. 1-1 if and only if T(x) = 0 has only the trivial solution (this means that A is linearly independent)
2. T maps Rn onto Rm if and only if the columns of A span Rm

Question 24

Invertibility

Answer 24

AC = I = CA

1. A and its inverse are invertible
2. Must be a square matrix, no free variables
3. (AB)-1 = B-1*A-1
4. Must be one to one and onto

Question 25

Vector Space

Answer 25

A non empty set V of objects which are subject to 10 axioms (on cheat sheet) 1. closed under addition 2. addition is commutative 3. addition is associative 4. closed under multiplication 5. has zero vector 6. an additive inverse exists 7. scalars distrivute 8. vecors distribute 9. associativity w scalars 10, scalar identity

Question 26

Subspace

Answer 26

1. zero vector in H 2. closed under addition 3. closed under multiplication

Question 27

Null Space

Answer 27

the solution set of A where x = 0 This is a subspace

Question 28

Column Space of A

Answer 28

The span of the column of A The pivot columns of a matrix A are the basis of Col (A)

Question 29

basis

Answer 29

a set of linearly independent vectors that span H