# Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number c_1. c_1cdotbegin{cases}x_1+x_2+x_3=05x_1-2x_2+2x_3=08x_1+x_2+5x_3=0end{cases}, X=c_1begin{pmatrix}43-7end{pmatrix}

Question
Matrices
Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$. $$c_1\cdot\begin{cases}x_1+x_2+x_3=0\\5x_1-2x_2+2x_3=0\\8x_1+x_2+5x_3=0\end{cases},\ X=c_1\begin{pmatrix}4\\3\\-7\end{pmatrix}$$

2021-01-17
Let's write
$$A=\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\ X=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$$
Matrix a, which has the dimensions $$3\times3$$, is the matrix of coefficients of the system, X, which has the dimensions $$3\times1$$ is the matrix of unknowns.
AX=0
$$\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$
If X is a solution of AX=0, then so is $$c_1X$$ for any constant c_1. Compute the product AX:
$$AX=\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\begin{bmatrix}4\\3\\-7\end{bmatrix}=\begin{bmatrix}1\cdot4+1\cdot3+1\cdot(-7)\\5\cdot4+(-2)\cdot3+2\cdot(-7)\\8\cdot4+1\cdot3+5\cdot(-7)\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$
Result:
$$\begin{bmatrix}1&1&1\\5&-2&2\\8&1&5\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$

### Relevant Questions

Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$
$$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$
$$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$
Consider a solution $$\displaystyle{x}_{{1}}$$ of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
a) If $$\displaystyle{x}_{{h}}$$ is a solution of the system Ax=0, then $$\displaystyle{x}_{{1}}+{x}_{{h}}$$ is a solution of the system Ax=b.
b) If $$\displaystyle{x}_{{2}}$$ is another solution of the system Ax=b, then $$\displaystyle{x}_{{2}}-{x}_{{1}}$$ is a solution of the system Ax=0
Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
$$\begin{cases}x_1+x_2-3x_3=-1\\-x_1+2x_2=1\\x_1-x_2+x_3=2\end{cases}$$
The row echelon form of a system of linear equations is given.
(a) Write the system of equations corresponding to the given matrix.
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
(b) Determine whether the system is consistent. If it is consistent, give the solution.
$$\begin{matrix}1 & 0 & 2 & -1 \\ 0 & 1 & -4 & -2\\0&0&0&0&0 \end{matrix}$$
The row echelon form of a system of linear equations is given.
(a) Write the system of equations corresponding to the given matrix.
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
(b) Determine whether the system is consistent. If it is consistent, give the solution.
$$\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}$$

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Find values of a and b such that the system of linear equations has no solution.
x+2y=3
ax+by=-9

Let B be a $$(4\times3)(4\times3)$$ matrix in reduced echelon form.

a) If B has three nonzero rows, then determine the form of B.

b) Suppose that a system of 4 linear equations in 2 unknowns has augmented matrix A, where A is a $$(4\times3)(4\times3)$$ matrix row equivalent to B.

Demonstrate that the system of equations is inconsistent.

$$x_1+2x_2-x_3=0$$
$$x_1+x_2+x_3=0$$
$$x_1+3x_2-3x_3=0$$
(a) Find a system of two linear equations in the variables x and y whose solution set is given by the parametric equations $$x = t$$ and y $$= 3- 2t.$$
(b) Find another parametric solution to the system in part (a) in which the parameter is s and $$y =s.$$