**Ax 0**always

**has**the trivial

**solution**, x

**0**. Nonzero vector

**solutions**are called

**nontrivial solutions**.

Just so, what is a nontrivial solution?

A **solution** or example that is not trivial. Often, **solutions** or examples involving the number zero are considered trivial. Nonzero **solutions** or examples are considered **nontrivial**. For example, the equation x + 5y = 0 has the trivial **solution** (0, 0). **Nontrivial solutions** include (5, –1) and (–2, 0.4).

Similarly, what is the solution set of the homogeneous system Ax 0? Thus, the **solution set** to **Ax** = **0** is Span{u,v,w}, or parametrically, x = ru + sv + tw where r,s,t ∈ R are parameters. Definition The **solution set** of a **homogeneous equation Ax** = **0** is called the kernel of A: ker A := {x ∈ Rn |**Ax** = **0**}.

Consequently, what it means for a system Ax 0 to have infinitely many solutions?

So if det (A) ≠ **0**, then **AX** = B has exactly one **solution**. If det (A) = **0**, then **AX** = B has **infinite solutions** or no **solutions**. The homogeneous **system** always has the trivial **solution** of X = **0**. **AX** = **0** has **infinitely many solutions**.

What is a homogeneous system?

A **system** of linear equations is **homogeneous** if all of the constant terms are zero: A **homogeneous system** is equivalent to a matrix equation of the form. where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.

###
What is the trivial solution for a system of homogeneous equations?

**Trivial Solution**to

**Homogeneous Systems**of

**Equations**. Suppose a

**homogeneous system**of linear

**equations**has n variables. The

**solution**x1=0 x 1 = 0 , x2=0 x 2 = 0 , …, xn=0 x n = 0 (i.e. x=0 ) is called the

**trivial solution**.

###
What is a trivial solution in linear algebra?

**trivial**if all its coordinates are 0, i. e. if it is the zero vector. In

**Linear Algebra**we are not interested in only finding one

**solution**to a system of

**linear**equations. In particular, homogeneous systems of equations (see above) are very important.

###
What is a null solution?

**null solution**(or as it’s more commonly called, the complementary

**solution**) is the

**solution**to the homogeneous equation. In this case, it is y = Ce

^{4t}. The particular

**solution**is a

**solution**to the nonhomogeneous equation.

###
What is a unique solution in linear equations?

**Unique Solution**to

**Linear Equations**

A system of **linear equations** ax + by + c = 0 and dx + ey + g = 0 will have a **unique solution** if the two lines represented by the **equations** ax + by + c = 0 and dx + ey + g = 0 intersect at a point. i.e., if the two lines are neither parallel nor coincident.

###
What do you mean by trivial solution?

**Trivial**. A

**solution**or example that is ridiculously simple and of little interest. Often,

**solutions**or examples involving the number 0

**are**considered

**trivial**. Nonzero

**solutions**or examples

**are**considered

**nontrivial**. For example, the equation x + 5y = 0 has the

**trivial solution**x = 0, y = 0.

###
What is the condition for non zero solution?

**non**–

**trivial solution**if and only if its determinant is

**non**–

**zero**. If this determinant is

**zero**, then the system has either no nontrivial

**solutions**or an infinite number of

**solutions**.

###
Can the solution set of Ax B be a plane through the origin?

**can**always write down the

**solution**; talking all variable to be a zero vector. If

**b**cannot equal zero,

**can Ax**=

**b be a plane through the origin**? No. The equation of a

**plane through the origin**has

**Ax**=

**b**and MUST = 0.

###
What is the rank of a matrix?

**rank of a matrix**is defined as (a) the maximum number of linearly independent column vectors in the

**matrix**or (b) the maximum number of linearly independent row vectors in the

**matrix**. Both definitions are equivalent. For an r x c

**matrix**, If r is less than c, then the maximum

**rank**of the

**matrix**is r.

###
What is a consistent system?

**system**has at least one solution, it is said to be

**consistent**. If a

**consistent system**has exactly one solution, it is independent . If a

**consistent system**has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.

###
What is linear homogeneous equation?

**homogeneous linear**differential

**equation**is a differential

**equation**in which every term is of the form y ( n ) p ( x ) y^{(n)}p(x) y(n)p(x) i.e. a derivative of y times a function of x. In fact, looking at the roots of this associated polynomial gives solutions to the differential

**equation**.

###
What is a free variable in linear algebra?

**Free**and Basic

**Variables**. A

**variable**is a basic

**variable**if it corresponds to a pivot column. Otherwise, the

**variable**is known as a

**free variable**. In order to determine which

**variables**are basic and which are

**free**, it is necessary to row reduce the augmented matrix to echelon form.

###
Why does the equation Ax B have a solution?

**Ax**means: it means we are multiplying the matrix A times the vector x. Solving

**Ax**=

**b**is the same as solving the system described by the augmented matrix [A|

**b**]. •

**Ax**=

**b**has a

**solution**if and only if

**b**is a linear combination of the columns of A.

###
Is every homogeneous linear system consistent?

**homogeneous system**must have at least one solution, . That

**is, every homogeneous system**of

**linear**equations is

**consistent**. Note: The solution is called the trivial solution to the

**homogeneous system**.

###
Is a system consistent if it has a free variable?

**consistent system**of linear equations must

**have a free variable**. false Reason:

**If**there are no

**free variables**then the

**system**can still be

**consistent**; it

**will have a**unique solution. Thus one column remains that always contributes a

**free variable**and infinitely many solutions.

###
What is Ax B when does Ax B has a unique solution?

**Ax**=

**b has a unique solution**if and only if the only

**solution**of

**Ax**= 0 is x = 0. Let A = [A1,A2,,An]. A rephrasing of this is (in the square case)

**Ax**=

**b has a unique solution**exactly when {A1,A2,,An} is a linearly independent set.

###
What is a in Ax B?

**Ax**, is the linear combination of the columns of A using the corresponding. entries in x as weights.

###
What kind of equation is Ax B 0?

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