Linear Algebra Flashcards
Solution
A list of values (s_1, s_2,…,s_n) that satisfy every equation of a linear system
Consistent
The solution set is nonempty
Inconsistent
The solution set is empty
Two linear systems are equivalent if
they share the same solution set
What are the options for the number of solutions a system can have?
1) Exactly one
2) Infinitely many
3) None
What does no solution look like in R2?
Two parallel lines
Two matrices are row equivalent if
There’s a sequence of elementary row operations that transform one matrix into another
In reduced row echelon for how many matrices are row equivalent?
Exactly one matrix
What is the rank of a matrix?
The number of pivot columns
What is the nullity of a matrix?
The number of free variable columns
If there are more variables than equations we…
Add a free variable
If there is a free variable then there are _______ solutions.
Infinitely many
Vectors u, v are equivalent if
u_1=v_1,…,u_n=v_n
Spanning set
The set of all linear combinations of a certain vector
Homogenous
The solution to the matrix equation Ax=b where b=0
Trivial solution
In the matrix equation Ax=b when x=0
The system Ax=0 has a nontrivial solution if and only if
it has at least one free variable
Linear independence
Each vector adds another dimension to the span
Linear dependence
When at least two vectors have the same span
Zero Matrix
An mxn matrix with all zero entries
Square Matrix
An nxn matrix
Main Diagonal
In a square matrix, it’s the values a_11, a_22,…,a_nn
Diagonal Matrix
Only the entries of the main diagonal are nonzero
Is the identity matrix a diagonal matrix?
Yes