Homogenous linear equation
all of the constant terms are 0
\_\_\_\_ = 0 \_\_\_\_ = 0 \_\_\_\_ = 0
A system of linear equations can only have how many solutions?
0, 1, ∞
If a system has at least ONE solution, what is it called?
consistent
If a system has NO solution, what is it called?
inconsistent
How can you determine the #dimensions in a system?
dimensions = #parameters
What are 3 ELEMENTARY row operations on a matrix?
- Multiply by non-zero constant
- Interchange two rows
- Add a constant times on row to another
True or False: A linear system whose equations are homogenous must be consistent?
True
What are the parameters necessary for a matrix to be in Reduced Row Echelon Form (RREF)?
- Leading one (non-zero rows)
- Lower leading one must be further to the RIGHT
- Rows of only 0 are at the BOTTOM
- Each column with a 1 must have ZERO everywhere else
What are the parameters necessary for a matrix to be in Row Echelon Form (REF)?
- Leading one (non-zero rows)
- Lower leading one must be further to the RIGHT
- Rows of only 0 are at the BOTTOM
DO NOT NEED 0 EVERYWHERE
What type of elimination do you do when converting a matrix into RREF?
Gauss-Jordan Elimination
What type of elimination do you do when converting a matrix into REF?
Gaussian Elimination
What do all homogenous systems do?
Pass through the origin (0,0)
What is a TRIVIAL solution of a homogenous system?
(0,0,0…0)
What is a NON-TRIVIAL solution of a homogenous system?
any other solution
NOT (0,0,0…0)
How do you know how many parameters a system will need?
parameters = # variables - # non-zero rows
What is a LEADING VARIABLE in a matrix?
leading 1s
What is a FREE VARIABLE in a matrix?
all other variables
NOT leading 1s
When a matrix needs to be parameterized how many solutions does it have?
∞
How many solutions would matrix with this last row have?
0 0 0 0 0 1
ZERO solutions
Does every matrix have a unique reduced row echelon (RREF) form?
Yes
Does every matrix have a unique row echelon (REF) form?
No
