Linear Algebra and Applications Flashcards

For these flashcards, I will be using David C Lay's Linear Algebra and Applications

A linear equation can be written in what form. Should the coefficients be real, complex or can they be both

The coefficients can be real OR complex. b can also be real or complex (Section 1.1)

Describe how many equations a linear equation can have

A system of linear equations has

1. No solution, or

2. Exactly one solution, or

3. Infinitely many solutions

(Section 1.1)

If m and n are positive integers, an m x n matrix has how many rows and columns

m rows and n columns (rows by columns) (Section 1.1)

What are the row operations used to simplify a linear system

1. Replacing one equation by the sum of itself and a multiple of another equation

2. Interchange two rows

3. Multiply one row by a nonzero constant

(Section 1.1)

What does it mean for a linear system to be consistent

It means the linear system has at least 1 solution (Section 1.1)

What does it mean for a linear system to be unique

It means the linear system has exactly one solution. It is the ONLY ONE (Section 1.1)

When is a matrix in echelon form (row echelon form)

A matrix is in echelon form if it has a staircase-like structure, where each leading entry is to the right of the leading entry of the previous row and all entries below a leading entry are zero. (Section 1.2)

When is a matrix in reduced echelon form (reduced row echelon form)

A matrix is in reduced echelon form if it is in echelon form, and additionally, each leading entry is 1 and is the only nonzero entry in its column. (Section 1.2)

A matrix with only one column is called what

A column vector (or just a vector) (Section 1.3)

What does it mean for vectors too be in R^n

Vectors are in R^n if they have n components in a column vector. (Section 1.3)

If a vector b is in the Span{v_1,…, v_p}, what are you also asking

If b can be represented as a linear combination of v_1,…,v_p. i.e, the image has a solution (x_1,x_2,…,x_p) (Section 1.3)

Based on the image, what are these statements

(Section 1.4)