If it can be written in the form of
?? = ?
Where ? is a ? x ? matrix and ? is the zero vector in ℝᵐ.
When is system of linear equations said to be homogenous?
It always has at least one solution,
namely, ? = ?. (The zero vector in ℝⁿ).
namely, ? = ?. (The zero vector in ℝⁿ).
How many solutions does a system like ?? = ? have?
The zero solution.
What is the trivial solution?
A nonzero vector ? that satisfies
?? = ?.
?? = ?.
What is a nontrivial solution?
The homogeneous equation ?? = ? has a nontrivial solution if and only if the equation has at least one free variable.
When combined with The Existence and Uniqueness theorem from 1.2, what fact does it immediately lead to?
x₃?.
What does the general solution vector look like?
It may be expressed explicitly as a Span {?₁, . . . , ??} for suitable vectors ?₁, . . . ??.
How may the solution set of a homogenous equation ?? = ? be expressed?
Then the solution set is Span {?}
What if the only solution is the zero vector?
Then the solution set is a line through the origin
What if the equation ?? = ? only has one free variable?
A plane through the origin.
What if the equation ?? = ? only has two or more free variables?
What is the implicit description of a plane?
By solving the implicit equation. Will yield an explicit description of the plane as the set spanned by ? and ?.
How do we find explicit description?
What is the parametric vector equation of the plane?
Whenever a solution set is described explicitly with vectors.
When is a solution in parametric vector form?
In parametric vector form as one vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogenous system.
How may the general solution be written of a non homogenous linear system that has many solutions?
?
as in
? = ? + ??
( ? in ℝ )
When writing the parametric vector form how may the free variable be represented?
? = ??
( ? in ℝ )
( ? in ℝ )
What does the solution set of ?? = ? have as a parametric vector equation
We can think of vector addition as translation.
How do you describe the solution set of ?? = ?, geometrically?
It is the equation of the line through ? parallel to ?.
The solution set of ?? = ? is a line through ? parallel to the solution set ?? = ?
What is the solution set of ?? = ? with respect to the solution set of ?? = ??
Suppose the equation ?? = ? is consistent for some given ? , and let ? be a solution. Then the solution set of ?? = ? is the set of all vectors of the form of ? = ? + ?_ℎ, where ?_ℎ is any solution of the homogenous equation ?? = ?.
What is THEOREM 6?
Only to an equation ?? = ? that has at least one NONzero solution ?. When ?? = ? has no solution, the solution set is empty.
When does THEOREM 6 apply?
1. Row reduce the augmented matrix to reduced echelon form.
2. Express each basic variable in terms of any free variables appearing in an equation.
3. Write a typical solution ? as a vector whose entries depend on the free variables, if any
4. Decompose ? into a linear combination of vectors with (numeric entries) using the free variables as parameters.
How do you write a solution set of a consistent system in parametric vector form?
When f(x) is equal to 0, or has infinitely many solutions
What is a homogeneous linear system?
When f(x) is not equal to 0.
What is a nonhomogenous linear system?
Because x = 0 in ℝⁿ solves ?? = ? in ℝᵐ.
Why are homogeneous linear systems always consistent?
The trivial solution of x = 0
If a homogenous linear system has only one solution, what is it?
Non-trivial.
If a homogenous linear system has infinitely may solutions, then what must they be?
When there is the presence of free variables which will imply infinitely many solutions, through the ability to change the value of the the free variable, it is possible to select one, that will yield a NONzero answer.
When does a homogenous system ?? = ? have a nontrivial solution?
Yes.
Can ?? = ? have infinitely many solutions without there being a row of zeros in the RREF?
