Question

obtain the row rank and parametrically represented solution of the system.
3x-y+3z=5
x+2y+2z-3w=-1
2x+5y+4z+2w=10

Answer 1

augment the coeff matrix

Answer 2

row rank = colA = dimA = number of pivot points

Answer 3

i didn't understand any of that.

Answer 4

then you really need to review your material, or ask more probing questions

Answer 5

what exactly is row rank?

Answer 6

make sure each equation HAS the same variables; whatevers missing put a 0 to it so that it is included

3x -1y+3z+0w=5
1x+2y+2z-3w=-1
2x+5y+4z+2w=10

strip off everything thats NOT a number


3 -1 3 0 5
1 2 2 -3 -1
2 5 4 2 10


now this is a coeff matrix that you can augment

Answer 7

row rank is the number of nonzero rows that you end up with

Answer 8

i was about to say this........

Answer 9

if we format this for the wolf, life gets a bit easier :)

rref{{3, -1, 3, 0, 5},{1, 2, 2, -3, -1},{2, 5, 4, 2, 10}}

Answer 10

ok i got that. what's next?

Answer 11

http://www.wolframalpha.com/input/?i=rref%7B%7B3%2C+-1%2C+3%2C+++0%2C+++5%7D%2C%7B1%2C++2%2C++2%2C+-3%2C++-1%7D%2C%7B2%2C++5%2C++4%2C+++2%2C++10%7D%7D

Answer 12

there are 3 nonzero rows; so row rank = 3

the last column of the augmented matrix is a constant vector

the next to last is the a free variable column

Answer 13

our parametric is based on these last 2 columns in this case; our constant vector and our free vectors

Answer 14

\[\vec x=\begin{pmatrix}x\\y\\z\\w \end{pmatrix}=\begin{pmatrix}109/3\\12\\-92/3\\0\end{pmatrix}+w\begin{pmatrix}-73/3\\-8\\65/3\\1 \end{pmatrix}\]

Answer 15

in this typa of questions, i was taught to suppose an arbitary constant and then calculate values of x,y,z,w for a particular value of arbitary constant.

Answer 16

yes, in this case the free variable is what you are calling an arbitrary constant

Answer 17

the augmented matrix produces our vectors for us; by augmenting with the "=" column vector to begin with we produce a particular value vector; and all thos coumns that dont stand on a "1" are called free variables ... arbitrary constants

Answer 18

it means that the rest of it can be defined using just those column vectors