# 311.1.5.12 Ax=0 in parametric vector form

• MHB
• karush
In summary,The problems in this section ask for solutions of equations in parametric vector form. One of the problems asks for a solution of an equation in three dimensions, while another asks for a solution of an equation in two dimensions. Both of these problems have solutions that can be found by solving the equations in terms of the given scalars and adding/subtracting the appropriate terms.
karush
Gold Member
MHB
$\tiny{1.5.12}$
Describe all solutions of $Ax=0$ in parametric vector form, where $A$ is row equivalent to the given matrix.
RREF
$A=\left[\begin{array}{rrrrrr} 1&5&2&-6&9& 0\\ 0&0&1&-7&4&-8\\ 0& 0& 0& 0& 0&1\\ 0& 0& 0& 0& 0&0 \end{array}\right] \sim \left[\begin{array}{rrrrrr} 1&5&0&8&1&0\\ 0&0&1&-7&4&-8\\ 0& 0& 0& 0& 0&1\\ 0& 0& 0& 0& 0&0 \end{array}\right]$
$x_1=-5x_2-8x_4-x_5$ $x_2$ free $x_3=7x_4-4x_5$ $x_4$ free $x_5\ free$x_6=0$solution\\$x_2\left[\begin{array}{rrrrrr}
-5\\1\\0\\0\\0\\0
\end{array}\right]
+x_4\left[\begin{array}{rrrrrr}
-8\\0\\7\\1\\0\\0
\end{array}\right]
+x_5\left[\begin{array}{rrrrrr}
-1\\0\\-4\\0\\1\\0
\end{array}\right]\$

ok this appears to be the answer but I still don't see how the origin is 0 or we have || planes

??This problem has nothing to do with "the origin" or "planes"!

ok i presume it is about parallel planes

Why? What was the exact statement of this problem and what makes you "presume" it is about parallel planes?

Country Boy said:
Why? What was the exact statement of this problem and what makes you "presume" it is about parallel planes?

#12

I see nothing there that says anything about "planes" or "parallel planes"!

ok so there is no possible graph of this

I have no idea what you are talking about! There is no mention of "planes" or "graphs" in this problem. Where are you getting this from? For problem 12, you have four equations in six unknowns. You could graph it- in 6 dimensions. The solution set is a two dimensional subspace of $$R^6$$.

But problem 9 has two equations in three dimensions: 3x- 6y+ 9z= 0 and -x+ 3y- 2z= 0. From the first equation, x= 2y- 3z. From the second equation, x= 3y- 2z. So x= 2y+ 3z= 3y- 2z. Add 2z to both sides and subtract 2y from both sides: 5z= y.
Then x= 3(5z)- 2z= 13z.

It solution space is one dimensional, the line in $$R^3$$, x= 13t, y= 5t, z= t.

## 1. What does "311.1.5.12" represent in the equation Ax=0 in parametric vector form?

The numbers "311.1.5.12" are most likely referring to the coefficients of the variables in the matrix A. Each number represents the value of a variable in the equation Ax=0.

## 2. What is the significance of "Ax=0" in parametric vector form?

In parametric vector form, the equation Ax=0 represents a system of linear equations where the solution is a zero vector. This means that the equations are consistent and have a unique solution.

## 3. How is the parametric vector form different from the standard form of a linear equation?

In the standard form of a linear equation, the variables are on one side of the equal sign and the constants are on the other side. In parametric vector form, the variables are grouped together in a matrix, and the constants are represented by the zero vector on the other side of the equal sign.

## 4. Can you solve for the values of x in the equation Ax=0 in parametric vector form?

Yes, it is possible to solve for the values of x in this equation. Since the solution is a zero vector, the values of x will be equal to zero. This means that the system of linear equations has a trivial solution where all variables are equal to zero.

## 5. How is the parametric vector form used in scientific research?

The parametric vector form is used in scientific research for modeling and solving systems of linear equations. It is also used in fields such as physics, engineering, and economics to represent and analyze linear relationships between variables.

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