Matrices Flashcards
(13 cards)
What is the trivial solution of a system of homogeneous linear equations?
The solution (0,0,0)
The trivial solution is the solution where all variables are equal to zero.
What is a non-trivial solution in the context of homogeneous linear equations?
Any other non-zero solution of the system
Non-trivial solutions exist when the system allows for solutions other than the trivial one.
What does it mean for a system of equations to be homogeneous?
It is a system where all equations equal zero
A homogeneous system can be represented as Ax = 0.
In matrices, a homogeneous system of linear equations can be written as____.
AX = B
Where B=0
How can a system of homogeneous linear equations be simplified?
By converting the matrix of coefficients to echelon form using elementary row operations
This process helps in finding solutions more easily.
What condition leads to only the trivial solution in a homogeneous system?
If the rank of A equals n or det(A) = 0
Here, n refers to the number of unknowns in the system.
What condition leads to both trivial and infinitely many non-trivial solutions?
If the rank of A is less than n or det(A) = 0
This indicates that the system has more unknowns than independent equations.
What is the matrix form of a system of homogeneous linear equations?
AX = 0
A is the matrix of coefficients, X is the vector of variables, and 0 is the zero vector.
What is the first step in solving a system of homogeneous linear equations?
Change the system of equations to the matrix form AX = 0
This step sets the stage for further manipulation and solution finding.
Fill in the blank: The solution to the homogeneous system 4x + 5y + 2z = 0 is represented in matrix form as _______.
AX = 0
True or False: A homogeneous system always has at least one solution.
True
A homogeneous system always has at least the trivial solution.
