True because heat breaks down the molecular structure and pressure reinforced the new contiguous configuration
Explanation:
The preparation of the year-end income statement for Fighting Okra Cooking Services is presented below:
Fighting Okra Cooking Services
Income statement
As on December 31
Revenue
Service revenue $78,500
Total revenues $78,500 (A)
Less: Expenses
Postage expense $1,500
Legal fees expense $2,600
Rent expense $21,000
Salaries expense $22,000
Supplies expense $20,000
Total expenses $67,100 (B)
Net income $11,400 (A- B)
On an organization's board of directors, inside directors <span>may be members of the firm; outside directors </span><span>are supposed to be elected from outside the firm.</span>
The board of directors is responsible for keeping the organization’s vision, mission, and strategic planning goals. Duties of boards include: <span>choosing the CEO, approving major policies, making major decisions, overseeing performance<span>, and serving as external advocate.</span></span>
Answer:
The vectors does not span R3 and only span a subspace of R3 which satisfies x+13y-3z=0
Explanation:
The vectors are given as
![v_1=\left[\begin{array}{c}-4&1&3\end{array}\right] \\v_2=\left[\begin{array}{c}-5&1&6\end{array}\right] \\v_3=\left[\begin{array}{c}6&0&2\end{array}\right]](https://tex.z-dn.net/?f=v_1%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-4%261%263%5Cend%7Barray%7D%5Cright%5D%20%5C%5Cv_2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-5%261%266%5Cend%7Barray%7D%5Cright%5D%20%5C%5Cv_3%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D6%260%262%5Cend%7Barray%7D%5Cright%5D)
Now if the vectors would span the 
, the rank of the consolidated matrix will be 3 if it is not 3 this indicates that the vectors does not span the .
So the matrix is given as
![M=\left[\begin{array}{ccc}v_1&v_2&v_3\end{array}\right] \\M=\left[\begin{array}{ccc}-4&5&6\\1&1&0\\3&6&2\end{array}\right]\\](https://tex.z-dn.net/?f=M%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dv_1%26v_2%26v_3%5Cend%7Barray%7D%5Cright%5D%20%5C%5CM%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-4%265%266%5C%5C1%261%260%5C%5C3%266%262%5Cend%7Barray%7D%5Cright%5D%5C%5C)
In order to calculate the rank, the matrix is reduced to the Row Echelon form as
![\approx \left[\begin{array}{ccc}-4&5&6\\ 0&\frac{9}{4}&\frac{3}{2}\\ 3&6&2\end{array}\right] R_2 \rightarrow R_2+\frac{R_1}{4}](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-4%265%266%5C%5C%200%26%5Cfrac%7B9%7D%7B4%7D%26%5Cfrac%7B3%7D%7B2%7D%5C%5C%203%266%262%5Cend%7Barray%7D%5Cright%5D%20R_2%20%5Crightarrow%20R_2%2B%5Cfrac%7BR_1%7D%7B4%7D)
![\approx \left[\begin{array}{ccc}-4&5&6\\ 0&\frac{9}{4}&\frac{3}{2}\\ 0&\frac{39}{4}&\frac{13}{2}\end{array}\right] R_3 \rightarrow R_3+\frac{3R_1}{4}\\](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-4%265%266%5C%5C%200%26%5Cfrac%7B9%7D%7B4%7D%26%5Cfrac%7B3%7D%7B2%7D%5C%5C%200%26%5Cfrac%7B39%7D%7B4%7D%26%5Cfrac%7B13%7D%7B2%7D%5Cend%7Barray%7D%5Cright%5D%20R_3%20%5Crightarrow%20R_3%2B%5Cfrac%7B3R_1%7D%7B4%7D%5C%5C)
![\approx \left[\begin{array}{ccc}-4&5&6\\ 0&\frac{39}{4}&\frac{13}{2\\ 0&\frac{9}{4}&\frac{3}{2}}\end{array}\right] R_2\:\leftrightarrow \:R_3](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-4%265%266%5C%5C%200%26%5Cfrac%7B39%7D%7B4%7D%26%5Cfrac%7B13%7D%7B2%5C%5C%200%26%5Cfrac%7B9%7D%7B4%7D%26%5Cfrac%7B3%7D%7B2%7D%7D%5Cend%7Barray%7D%5Cright%5D%20R_2%5C%3A%5Cleftrightarrow%20%5C%3AR_3)
![\approx \left[\begin{array}{ccc}-4&5&6\\ 0&\frac{39}{4}&\frac{13}{2}\\ 0&0&0\end{array}\right] R_3 \rightarrow R_3-\frac{3R_2}{13}\\](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-4%265%266%5C%5C%200%26%5Cfrac%7B39%7D%7B4%7D%26%5Cfrac%7B13%7D%7B2%7D%5C%5C%200%260%260%5Cend%7Barray%7D%5Cright%5D%20R_3%20%5Crightarrow%20R_3-%5Cfrac%7B3R_2%7D%7B13%7D%5C%5C)
As the Rank is given as number of non-zero rows in the Row echelon form which are 2 so the rank is 2.
Thus this indicates that the vectors does not span
<em>Now for any vector the corresponding equation is formulated by using the combined matrix which is given as for any arbitrary vector and the coordinate as </em>
<em />![v=\left[\begin{array}{c}x&y&z\end{array}\right]](https://tex.z-dn.net/?f=v%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx%26y%26z%5Cend%7Barray%7D%5Cright%5D)
<em />
<em />![c=\left[\begin{array}{c}c_1&c_2&c_3\end{array}\right]](https://tex.z-dn.net/?f=c%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dc_1%26c_2%26c_3%5Cend%7Barray%7D%5Cright%5D)
<em />
<em />
<em />
<em />![\left[\begin{array}{ccc}-4&5&6\\1&1&0\\3&6&2\end{array}\right]\left[\begin{array}{c}c_1&c_2&c_3\end{array}\right]=\left[\begin{array}{c}x&y&z\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-4%265%266%5C%5C1%261%260%5C%5C3%266%262%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dc_1%26c_2%26c_3%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx%26y%26z%5Cend%7Barray%7D%5Cright%5D)
<em />
<em />![M=\left[\begin{array}{ccccc}v_1&v_2&v_3& | &v\end{array}\right] \\M=\left[\begin{array}{ccccc}-4&5&6&|&x\\1&1&0&|&y\\3&6&2&|&z\end{array}\right]\\](https://tex.z-dn.net/?f=M%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7Dv_1%26v_2%26v_3%26%20%7C%20%26v%5Cend%7Barray%7D%5Cright%5D%20%5C%5CM%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D-4%265%266%26%7C%26x%5C%5C1%261%260%26%7C%26y%5C%5C3%266%262%26%7C%26z%5Cend%7Barray%7D%5Cright%5D%5C%5C)
<em />
Now converting the combined matrix as
![\approx \left[\begin{array}{ccccc}-4&5&6&|&x\\ 0&\frac{9}{4}&\frac{3}{2}&|&\frac{4y+x}{4}\\ 3&6&2&|&z\end{array}\right] R_2 \rightarrow R_2+\frac{R_1}{4}\\](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D-4%265%266%26%7C%26x%5C%5C%200%26%5Cfrac%7B9%7D%7B4%7D%26%5Cfrac%7B3%7D%7B2%7D%26%7C%26%5Cfrac%7B4y%2Bx%7D%7B4%7D%5C%5C%203%266%262%26%7C%26z%5Cend%7Barray%7D%5Cright%5D%20R_2%20%5Crightarrow%20R_2%2B%5Cfrac%7BR_1%7D%7B4%7D%5C%5C)
![\approx \left[\begin{array}{ccccc}-4&5&6&|&x\\ 0&\frac{9}{4}&\frac{3}{2}&|&\frac{4y+x}{4}\\ 0&\frac{39}{4}&\frac{13}{2}&|&\frac{4z+3x}{4}\end{array}\right] R_3 \rightarrow R_3+\frac{3R_1}{4}\\](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D-4%265%266%26%7C%26x%5C%5C%200%26%5Cfrac%7B9%7D%7B4%7D%26%5Cfrac%7B3%7D%7B2%7D%26%7C%26%5Cfrac%7B4y%2Bx%7D%7B4%7D%5C%5C%200%26%5Cfrac%7B39%7D%7B4%7D%26%5Cfrac%7B13%7D%7B2%7D%26%7C%26%5Cfrac%7B4z%2B3x%7D%7B4%7D%5Cend%7Barray%7D%5Cright%5D%20R_3%20%5Crightarrow%20R_3%2B%5Cfrac%7B3R_1%7D%7B4%7D%5C%5C)
![\approx \left[\begin{array}{ccccc}-4&5&6&|&x\\ 0&\frac{39}{4}&\frac{13}{2}&|&\frac{4z+3x}{4}\\ 0&\frac{9}{4}&\frac{3}{2}&|&\frac{4y+x}{4}\end{array}\right] R_3 \leftrightarrow R_2\\](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D-4%265%266%26%7C%26x%5C%5C%200%26%5Cfrac%7B39%7D%7B4%7D%26%5Cfrac%7B13%7D%7B2%7D%26%7C%26%5Cfrac%7B4z%2B3x%7D%7B4%7D%5C%5C%200%26%5Cfrac%7B9%7D%7B4%7D%26%5Cfrac%7B3%7D%7B2%7D%26%7C%26%5Cfrac%7B4y%2Bx%7D%7B4%7D%5Cend%7Barray%7D%5Cright%5D%20R_3%20%5Cleftrightarrow%20R_2%5C%5C)
![\approx \left[\begin{array}{ccccc}-4&5&6&|&x\\ 0&\frac{39}{4}&\frac{13}{2}&|&\frac{4z+3x}{4}\\ 0&0&0&|&\frac{13y+x-3z}{13}\end{array}\right] R_3 \rightarrow R_3-\frac{3R_2}{13}\\](https://tex.z-dn.net/?f=%5Capprox%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D-4%265%266%26%7C%26x%5C%5C%200%26%5Cfrac%7B39%7D%7B4%7D%26%5Cfrac%7B13%7D%7B2%7D%26%7C%26%5Cfrac%7B4z%2B3x%7D%7B4%7D%5C%5C%200%260%260%26%7C%26%5Cfrac%7B13y%2Bx-3z%7D%7B13%7D%5Cend%7Barray%7D%5Cright%5D%20R_3%20%5Crightarrow%20R_3-%5Cfrac%7B3R_2%7D%7B13%7D%5C%5C)
From this it is seen that whatever the values of the coordinates does not effect the value of the plane with equation as
So it is verified that the subspace of R3 such that it satisfies x+13y-3z=0 consists of all vectors.
<em />
Preserved remains of organisms are known as fossils.
Fossils are generally <u>remains of ancient dead living organisms, including plants and animals</u>. Fossils can be in the form of bones, shells, or even an animal’s imprint.
These things help Charles Darwin in proving his theory of evolution, which states that heritable characteristics in living organisms change over several generations.
