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Transformations

100

Given (2,5) in R² + (8.3) in R² what does that equal?

2 8

5 3

200

Given v1 = (3, 2, 6) in R³, v2 = (6,2,7) in R³, and v3 = (3,7) in R²

^{Perform (v1+v2) * 6(v3)}

work the problem out

300

Is x1 = [ 1 ] + x2 = [ 3 ] = [ 7 ]

            [ 6 ]            [ 8 ]    [ 6 ] 

            [ 2 ]            [ 0 ]    [ 10 ]               

Reword the question in terms of Span

work problem out

400

what is a linear transformation (definition)

T is a function that maps vectors in Rⁿ to vectors in R^mand has the linearity properties

for any v, v in Rⁿ and real numbers:

i: T(U + V) = T(U)+T(V)

ii: T(CU) = CT(U)

500

take matrix A = 1 -5, 3 5, 2 4, and show how this is a Transformation from R²->R³

See 1.8 bottom

Reduced Row Echelon Form

100

Solve this matrix:137, 437, 462, convert to rref

3x3 Identity matrix

200

Given the matrix 137, 437, 86(14), convert to rref

100 01 7/3 000

300

Find RREF of 0201, 4230, 1122, 2101

Identity

400

RREF of 023, 423

1 0 0

0 1 1.5

500

6 2 3, 6 2 3, 6 2 1 in RREF

1 .33 .33

0 0 0

0 0 -2

Linear Independance

100

What does it mean to be linear independant?

Only has a trivial solution

200

THM: Ax = b has how many solutions

1 AT MOSY

300

Ax = 0 has how many soltuons

only a trivial solution

400

How many solutions does ax = b have where A = 133, 174, b = 21

idk

500

How many solutions does ax = b have where A = 184, 394, b = 59

work it out

Definitions / Theorems

100

What is a linear equation (definition) 

and

 What is a system of linear equations (definition)

 

A linear equation in (x1, x2, x3, ... xn) is an equation that can be written in form a1x1+a2x2+a3x3 +...+ anxn = b

for some real or complex numbers, a1,a2,a3 ... an = b

A system of linear equations is a collection of linear equations.

200

What does consistency and inconsistency mean in terms of matricies?

has solutions or does not have solutions

300

What is the Existence and uniqueness theorem

A linear system is consistent if and only if an echelon form of the augmented matrix has no row of form [0...0 b] 

If the system is consistent the solution is unique or there are infinitely many solutions

400

What is a pivot and a pivot column?

What is every other variable other than a pivot called?

The 1's in the identity matrix

basic variables

500

If A has a pivot in every row what else do we know is true?

The columns of A span R^m

V(b) in R^m, aV(x) = V(b) is consistent,

V(b) = b as a vector

Row Transformations

100

Given matrix : 5542, 1137, 9438, 2388, what should be multiplied by the first row to eliminate the first item of the second row?

1/5

200

When reducing a Matrix, what happens when you are left with 0 = 1?

The matrix is inconsistent.

300

Given augmented matrix: 00804, 59233, 03874, which order should the rows be put in for easiest row transforming?

2, 3, 1

400

What are the basic row opperations

cmon

500

Go through every step in reducing this matrix to REF: 174. 831. 002

walk through it with them and they should come to an answer