1.3 Vector Equations

T OR F: A matrix with one column is a column vector, or simply, a vector

TRUE

A matrix consisting of a single column is classified as a column vector, which represents a vector in a multi-dimensional space.

What is vector addition?

Two vectors of the same size (same number of rows) are added COMPONENT wise

So, you take the first entry in both and add together to get the new entry of the new column and repeat this for all entries, resulting in a new vector that retains the same dimensions.

What is scalar multiplication?

Involves a column vector and a real number ‘c’

Each entry in the vector is multiplied by c to produce a new vector.

What is the zero-vector?

A special vector in which all its components are zero. It serves as the additive identity in vector addition, meaning that adding the zero-vector to any vector gives the same vector.

What can the sum vector u + vector v represent in terms of a parallelogram?

The sum u + v is the fourth vertex of the parallelogram whose other vertices are 0 vector (origin), vector u, vector v

T OR F: Scalar multiplication “stretches” the line through the origin to the vector u

TRUE

when the scalar is greater than 1 and shrinks when it is between 0 and 1. If the scalar is negative, it also reverses the direction of the vector.

What linear system does the vector equation x₁a₁ + x₂a₂ + … + x_na_n = vector b have the same solution set as (augmented matrix answer)

the linear system represented by the augmented matrix [a1, a2, …, a_n | b].

Define a spanning set

A spanning set is a set of vectors that can be combined through linear combinations to produce any vector in a given vector space. It ensures that the entirety of the vector space is covered.

What is determining if b E span {v₁, …, v_p} (is b in the plane) equivalent to?

determining if the linear system has a solution.