# 311.1.5.5 homogeneous systems in parametric vector form.

• MHB
• karush
In summary, the given homogeneous system can be written in parametric vector form as:$\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -2 \\ -4 \\ 0 \end{bmatrix} x_1 + \begin{bmatrix} 2 \\ -4 \\ 3 \end{bmatrix} x_2 + \begin{bmatrix} 4 \\ -8 \\ -3 \end{bmatrix} x_3$, where the null space is represented by the zero vector. This solution can be checked using an online calculator.
karush
Gold Member
MHB
Write the solution set of the given homogeneous systems in parametric vector form.
$\begin{array}{rrrr} -2x_1& +2x_2& +4x_3& =0\\ -4x_1& -4x_2& -8x_3& =0\\ &-3x_2& -3x_3& =0 \end{array}\implies \left[\begin{array}{rrrr} x_1\\x_2\\x_3 \end{array}\right] =\left[\begin{array}{rrrr} -2\\-4\\\color{red}{0} \end{array}\right]x_1 +\left[\begin{array}{rrrr} 2\\-4\\-3 \end{array}\right]x_2 +\left[\begin{array}{rrrr} 4\\-8\\-3 \end{array}\right]x_3$
red is a null space

ok its looks straight forward but still ? typos etc
is there an online calculator to check these
no book answer on this one

karush said:
Write the solution set of the given homogeneous systems in parametric vector form.
$\begin{array}{rrrr} -2x_1& +2x_2& +4x_3& =0\\ -4x_1& -4x_2& -8x_3& =0\\ &-3x_2& -3x_3& =0 \end{array}\implies \left[\begin{array}{rrrr} x_1\\x_2\\x_3 \end{array}\right] =\left[\begin{array}{rrrr} -2\\-4\\\color{red}{0} \end{array}\right]x_1 +\left[\begin{array}{rrrr} 2\\-4\\-3 \end{array}\right]x_2 +\left[\begin{array}{rrrr} 4\\-8\\-3 \end{array}\right]x_3$
red is a null space

ok its looks straight forward but still ? typos etc
is there an online calculator to check these
no book answer on this one
No. The sum is not equal to "$\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}$. It is equal to "$\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$".
I also do not understand why you have written the "0" in red and called it a "null space". It is simply the number 0.

This is $\begin{bmatrix} -2 \\ -4 \\ 0 \end{bmatrix} x_1+ \begin{bmatrix} 2 \\ -4 \\ 3 \end{bmatrix} x_2+ \begin{bmatrix} 4 \\ -8 \\ -3 \end{bmatrix}x_3= \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}$.

ok i tried to follow a hand written example in saw on Google images

## 1. What is a homogeneous system in parametric vector form?

A homogeneous system in parametric vector form is a system of equations where the variables are represented as vectors and the coefficients are represented as parameters. This form is commonly used in linear algebra and is useful for solving systems of equations with multiple variables.

## 2. How is a homogeneous system in parametric vector form different from a standard system of equations?

A homogeneous system in parametric vector form is different from a standard system of equations because it uses vectors and parameters instead of individual variables and coefficients. This form allows for more efficient and concise representation of systems with multiple variables.

## 3. What is the purpose of using parametric vector form in homogeneous systems?

The purpose of using parametric vector form in homogeneous systems is to simplify and generalize the process of solving systems of equations. This form allows for a more streamlined approach to solving systems with multiple variables and can also reveal important properties of the system.

## 4. How do you solve a homogeneous system in parametric vector form?

To solve a homogeneous system in parametric vector form, you can use techniques such as Gaussian elimination or matrix operations. You can also use the properties of homogeneous systems to simplify the equations and find a solution. It is important to remember that a homogeneous system may have infinitely many solutions or no solutions at all.

## 5. What are some real-world applications of homogeneous systems in parametric vector form?

Homogeneous systems in parametric vector form have various applications in fields such as engineering, physics, and computer science. They are commonly used in solving problems related to linear transformations, optimization, and control systems. They can also be used to model and analyze complex systems with multiple variables and parameters.

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