Linear Algebra Test 1 (sec 1.1 - sec 3.1)

These are all the definitions from section 1.1 to section 3.1. They are explained as if a 5 year old is reading them so they are simple to wrap your head around.

Vector (n-dimensional)

A list of numbers written in a column, like a tall stack.

Vector equality

Two vectors are the same if all their numbers match.

Vector addition

Add vectors by adding the numbers in the same spot.

Scalar multiplication

Multiply a vector by a number by multiplying each piece.

Linear combination

Mix some vectors together using multiplication and addition.

Standard coordinates

Special “basic” vectors that point along each axis (like the building blocks of all vectors).

Norm (length)

How long a vector is, like measuring with a ruler.

Distance

How far two vectors are from each other.

Angle between vectors

How much you’d have to “turn” one vector to line it up with another.

Orthogonal

Two vectors are at right angles (90°), like the corner of a box.

System of equations

A bunch of linear equations that share the same variables.

Solution

Numbers that make the equation true.

Solution set

All the possible answers that work.

Consistent system

A system with at least one answer.

Inconsistent system

A system with no answers.

Coefficient matrix

A grid that just holds the equation’s coefficients (the number in front of variables).

Augmented matrix

The coefficient matrix but with the answer numbers added at the end.

Matrix

Just a big rectangle (grid) of numbers.

Matrix-vector multiplication

Combining a matrix and a vector to make a new vector.

Reduced Row Echelon Form (RREF)

Like REF, but the leading numbers are 1, and each column has only that 1.

Leading variables

Variables that match with pivot columns.

Free variables

Variables that don’t match with pivots — they can be anything.

Homogeneous system

A system where the equations all equal zero.

Linearly dependent

Some vectors can be made from others in the set.

Linearly independent

None of the vectors can be made from the others.

Column space

All vectors you can make by combining the columns of a matrix.

Scalar multiplication (matrix)

Multiply every number in the matrix by the same number.

Matrix addition

Add two matrices by adding matching numbers.

Zero matrix

A matrix made of all zeros.

Transpose

Flip a matrix over its diagonal (rows become columns).

Identity matrix

A square matrix with 1’s down the diagonal, 0’s everywhere else. It’s like “1” for matrices.

Matrix powers

Multiply a matrix by itself over and over.

Dot product

Multiply matching numbers of two vectors, then add them up.

Linear equation

An equation where numbers and variables are only added, multiplied, or scaled (no powers or weird stuff).

Row Echelon Form (REF)

A matrix arranged so each leading number moves right as you go down (like stairs).

Span

All the vectors you can make by mixing a set of vectors.

Equal matrices

Two matrices are the same if every number in the grid matches.