MATH0520 - LE1-4, EV1, EV2

matrix

an 𝑚 ×𝑛 array of real numbers with 𝑚 rows and 𝑛 columns

column vectors

set notation for matrices

</span> mathematical notation used to represent a matrix's structure, typically involving brackets and commas to separate elements within rows and columns </span>

Euclidean vector

an ordered list of real numbers

component

each number in the list/euclidean vector

linear equation

equation of the variables 𝑥𝑖 of the form

solution

linear equation is a Euclidean vector that satisfies

𝑎1⁢𝑠1+𝑎2⁢𝑠2+⋯+𝑎𝑛⁢𝑠𝑛=𝑏 (that is, a Euclidean vector whose components can be plugged into the equation).

system of linear equations or a linear system

is a collection of one or more linear equations

solution set

is a solution to all equations in the system

consistent

if its solution set is non-empty (that is, there exists a solution for the system

inconsistent

if its solution set is empty (that is, there is no solution for the system)

consistent with one solution

solution set for linear system that is contains a single vector (one unique solution)

consistent with infinite solution

solution set for linear system that contains infinitely many vectors (infinite solution)

inconsistent

solution set for linear system that is empty set, denoted by either {} or ∅, no solution

augmented matrix

the 𝑚 ×𝑛 matrix of its coefficients augmented with the 𝑚 constant values as a final column

coefficient matrix

𝑚 ×𝑛 array consisting of the coefficients of the linear system (and ignore its constant terms)

Vector equation

an equation written as a linear combination of column vectors with unknown scalar variables set equal to a target result vector

equivalent

Two systems of linear equations are that have the same solution set (denote with ∼)

three row operations produce equivalent augmented matrices

1. Swap two rows, for example, 𝑅1 ↔𝑅2

2. Multiply a row by a nonzero constant, for example, 2⁢𝑅1 →𝑅1

3. Add a constant multiple of one row to another row, for example, 𝑅2 −4⁢𝑅1 →𝑅2

reduced row echelon form (RREF)

1. The leftmost nonzero term of each row is 1. We call these terms pivots.

2. Each pivot is to the right of every higher pivot.

3. Each term that is either above or below a pivot is 0.

4. All zero rows (rows whose terms are all 0) are at the bottom of the matrix

Gauss-Jordan elimination algorithm

1. Ignoring any rows that already have marked pivots, identify the leftmost column with a nonzero entry.

2. Use row operations to obtain a pivot of value 1 in the topmost row that does not already have a marked pivot.

3. Mark this pivot, then use row operations to change all values above and below the marked pivot to 0.

4. Repeat these steps until the matrix is in RREF.

inconsistent linear system (RREF)

If the linear system given by RREF⁡(𝐴) includes the contradiction

0=1, that is, the RREF matrix includes the row [0⋯01]

consistent linear system with one solution (RREF)

If the linear system given by RREF⁡(𝐴) sets every variable of the system to a specific value; that is we have:

𝑥1=𝑠1

𝑥2=𝑠2⋮

𝑥𝑛=𝑠𝑛

(with some possible extra 0 = 0 equations), then

consistent with infinitely-many different solutions (RREF)

not include a 0 =1 contradiction while at least one variable is not required to equal a specific value.

bound variables

pivot columns in an augmented matrix

free variables

a variable that does not correspond to a pivot column in row-echelon form, meaning it can take any real number value

scalar

The real number that is used to multiply a vector

vector spaces

a collection of objects called vectors that can be added together and scaled by numbers called scalar (e.g. polynomials, matrices)

linear combination

a set of vectors {v1,v2,…,vn} is given by c1v1 +c2⁢v2 +… +𝑐𝑛⁢𝑣𝑛 for any choice of scalar multiples 𝑐1,𝑐2,…,𝑐𝑛.

span of a set of vectors

is the collection of all linear combinations of that set

span in R^m

𝑚 vectors are required to span ℝ^𝑚

{v1,…,vn} spans all of ℝ^m

is consistent for every vector meaning the vector is in R^m (RREF of the vector equation’s coefficient matrix has all pivot rows)

{v1,…,vn} fails to span all of ℝ^𝑚

when the vector equation

𝑥1⁢⃗𝑣1+⋯+𝑥𝑛⁢⃗𝑣𝑛=⃗𝑤

is inconsistent for some vector ⃗𝑤 ∈ ℝ^𝑚 (RREF of the vector equation’s coefficient matrix has at least one non-pivot row (a row of zeroes)