Module 1 GT DE Math

When is a matrix consistent?

When it has 1 or more solutions

When is a matrix inconsistent?

When it has no solutions

What is row equivalence?

If a sequence of row operations transforms one matrix into another (those two matrices are row equivalent)

What happens to the solution set of one matrix when it is transformed into another row equivalent matrix?

the solution set stays the same

What are the requirements for echelon form?

• All zero rows (if any) are at the bottom

• First non-zero entry of a row (if any) is to the right of any leading entries in the above rows

• All entries below a leading entry are zero

What are the requirements for RREF?

• Must already be in echelon form

• All leading entries (if any) must be equal to 1

• Leading entries are the only non-zero entry in their column

What is a pivot position?

location in a matrix corresponding to a leading 1 in the RREF

What is a pivot column?

column that contains a pivot position

What are basic variables?

variables that correspond to a pivot

What are free variables?

any variable that is not basic

Do matrices alone have basic/free variables?

no, only systems do

When is a linear system consistent?

if and only if the last column of the augmented matrix doesnt have a pivot

it cannot have a \begin{pmatrix} 0 & 0 & 0… & 0 | 1 \end{pmatrix}

When does a linear system have a unique solution?

if and only if there are no free variables

When does a linear system have infinitely many solutions?

if it has free variables

What does \mathbb{R}^n mean?

an n-dimensional space and the vectors in it

What is the definition of a linear combination?

Given…

\overrightarrow{v_1}, \overrightarrow{v_2}, … , \overrightarrow{v_p} \in \mathbb{R}^n

and scalars

c_1,c_2,…c_p

the vector \overrightarrow{y} = c_1\overrightarrow{v_1} + c_2\overrightarrow{v_2} + … + c_p\overrightarrow{v_p}

What is the definition for a span of a set of vectors?

Given vectors

\overrightarrow{v_1},\overrightarrow{v_2},…,\overrightarrow{v_p} \in \mathbb{R}^n

and scalars

c_1,c_2,…,c_p

The set of all linear combinations is called the span of \overrightarrow{v_1},\overrightarrow{v_2},…,\overrightarrow{v_p}

True or false: any 2 non-parallel vectors in \mathbb{R}³ \mathbb{R}³ span a plane that passes through the origin. Any vector in that plan is in the span.

True

What does \mathbb{R}^{m*n} mean?

set of real-valued matrices with m-rows and n-columns

What is the definition of a matrix-vector product?

If A \in \mathbb{R}^n has columns,

\overrightarrow{a_1},…\overrightarrow{a_n} and \overrightarrow{x} \in \mathbb{R}^n , then

the matrix vector product A\overrightarrow{x} is a linear combination of the columns of A

Click to see a visual representation of a matrix-vector product

A\overrightarrow{x} = \begin{pmatrix} | & | & … & |\\\overrightarrow{a_1} & \overrightarrow{a_2} & … & \overrightarrow{a_n}\\| & | & … & | \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ | \\ x_n \end{pmatrix} = x_1\overrightarrow{a_1} + x_2\overrightarrow{a_2} + … + x_n\overrightarrow{a_n}

A is an m*n matrix with columns \overrightarrow{a_1},…,\overrightarrow{a_n} and \overrightarrow{x} \in \mathbb{R}^n. Describe the solutions to A\overrightarrow{x} = \overrightarrow{b} and x_1\overrightarrow{a_1} +…+x_n\overrightarrow{a_n} = \overrightarrow{b} and \begin{bmatrix} \overrightarrow{a_1} & \overrightarrow{a_2} & … & \overrightarrow{a_n} & | & \overrightarrow{b} \end{bmatrix}

They will all have the same solutions

When does the linear combination A\overrightarrow{x}=\overrightarrow{b} have a solution?

If and only if \overrightarrow{b} is a linear combination of the columns of A