linear alg

E

is an element of

span

set of all linear combinations of those vectors

mxn

m rows and n columns in the matrix

dimension

if a vector has n coordinates, its in dimension n

inconsistent

a linear system is inconsistent if a row of 0s equals a number

elementary row operations

1. addition of row and scalar of another
   2. swapping rows
   3. scalar of row

homogeneous matrix

Ax=0 when the solution side of the augmented is all 0s or it’s a coefficient matrix

in-homogeneous

Ax=b
where b is not = to 0

p vectors

cannot span R^m if n<m

equivalent

two matrices have the same solution set

If A is an mxn matrix, then:

1. for each b in R^m, the equation Ax=b has a solution
   2. each b in R^m is a linear combo. of the columns of A
   3. the columns of A span R^m
   4. A has a pivot position in every row (A is a coefficient matrix - not augmented)

Linear Equation

An equation that can be written as a₁x₁ + a_2×2 + a₃x₃ +…+ a_nx_n, where a₁,a₂,…a_n, b are real or complex numbers

Linear equations

A collection of two or more linear equations using the same variables

Solutions set

The set of all possible solutions to a system

Graphs

REF vs. RREF

REF:
- all non-zero rows are above all all-zero rows
- each leading entry of a row is in a column to the right of the LE of the row above it
all entries in a column below a LE are zeros
RREF:
- REF
- the LE in each non-zero row is 1
- each leading 1 is he only non-zero entry in the column

Pivot Position:
Pivot Column:
Pivot:

• corresponds to leading entry

• the column that contains the pivot

• nonzero number in pivot position used to create zeros with row operations

Free variables

can take on any value. once you choose a a value for your free variable, it will determine the value of the other (basic) variables

Vector

An ordered list of numbers

Column vector

a vector with only one column, we often use these for ordered pairs, triples, etc.

Vectors in R²

The set of all vectors with 2 entries. R > real numbers 2> number of entries. THis is the set of all points in a plane

Operations with vectors

Scalar - multiply by a constant
Addition - add corresponding values
Multiplication - have to pay attention to the dimensions. to multiply, you need JUST 1 row x how many columns

Vectors in Rⁿ

If n E R, then Rⁿ is the collection of all lists of ordered n-tuples (written as nx1 column matrix) of n real numbers written as nx1 column matrices

Algebraic properties of R²

Linear combinations

The vector defined by y=c₁x₁ + c₂×+….+ c_nx_n, where c_i are scalars and v_i are vectors, is called a linear combo. of v_i, v₂, v₃ …v_n with weight of c₁, c₂,…

A vector equation

a₁x₁ + a_2×2 + a₃x₃ +…+ a_nx_n has the same solution set as the linear system whose augmented matrix is [a_i, a₂, a₃,…]. Therefore, a vector equation only has a solution if the system is consistent.

If v_i, v₂, v₃ …v_p are in Rⁿ

then the set of linear combo. is denoted span{v_i, v₂, v₃ …v_n} and is called the subset of Rⁿ spanned. Span is essentially all the vectors that can be written in the form c₁x₁ + c₂×+….+ c_nx_p = b with c_i scalars

Matrix equation Ax=b

If A is an mxn matrix with columns a₁, a₂,…, a_n and if x E Rⁿ is the linear combo. of the columns of A using the corresponding entries in x as weights.

Same but different

Row-vector rule for computing Ax

If Ax is defined, the i^th entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x. Basically, taking the whole row and multiply it by the column and it’ll give you the sol. [²₃⁴₅] and vector [¹₂] you should do 1×2 + 2×4 and then 3 × 1 time 5×2 and that’ll give [¹⁰₁₃]. Note the columns of the A and the rows of the x (vector) have to be the same number

For homogeneous: a linear system can be written in the form Ax = 0

Trivial solution: x = 0 (0 vector is always a sol.)
Non-trivial solution: x does NOT = 0 ( we want to solve for some value of vector x that makes the matrix x vector = 0) [²_-1^-2₁] x [³₃] = still equil [⁰₀] even though x is not 0,

Ax = 0 MUST have at least 1 free variable

Parametric vector form/equation

you can write x as some constant times a vector (x = tv), vector x = [x₁, x₂, x₃] x [(4/3)X₃, 0, ,x₃] = x₃ [4/3 , 0 , 1].

Needs to be in RREF and can have mult. free variables o

Translating homogenous solutions

Solutions of Ax = b are translations. to solutions of Ax = 0 where Ax = 0 line is through origin at 0,0. x = p + tv where p is the translation and the tv is the same solution from the homogeneous (t is variable and v is vector)

Parametric equation

all 0s in solution column for augmented matrix

independent vector

A set of vectors is linearly independent if none of them can be expressed as a linear combination of the others. This means each vector provides new, non-redundant directional information, and the only way to combine them to equal the zero vector is by multiplying all vectors by zero