Let's consider an arbitrary 2x2 matrix as an example,
The columns of
are linearly independent if and only if the column vectors
are linearly independent.
This is the case if the only way we can make a linear combination of
reduce to the zero vector is to multiply the vectors by 0; that is,
only by letting
.
A more concrete example: suppose
Here,
and
. Notice that we can get the zero vector by taking
and
:
so the columns of
are not linearly independent, or linearly dependent.
First let's talk about the blue line.
You can see its rising so its slope is certainly positive. But by how much is it rising? You can observe that each unit it rises it goes 1 forward and 1 up so its slope is the ratio of 1 up and 1 forward which is just 1.
We have thusly,
Now look at where blue line intercepts y-axis, -1. That is our n.
So the blue line has the equation of,
Next the black lines. The black lines are axes so their equations are a bit different.
First let's deal with x-axis, does it have slope? Yes but it is 0. The x-axis is still, not rising nor falling. Where does x-axis intercept y-axis? At 0. So the equation would be,
Now we have y-axis. Does y axis have a slope? Yes but it is 
. The y-axis rises infinitely in no run. Where does it intercept y-axis? Everywhere! So what should the equation be? What if we ask where does y-axis intercept x-axis and write its equation in terms of x. Y-axis intercepts x-axis at 0 which means its equation is,
That is, every point of a form 
lies on y-axis.
Hope this helps :)
Answer:
The 6 oz bottle
Step-by-step explanation:
When you simplify it to see how much per ounce each costs, the 6-oz has the better price.
2.29/6 = 0.381666....
10.19/15 = 0.6793....
the second bottle cost more per ounce, so the 6 oz bottle has the better deal
*plz brainlist
Answer:
58240
Step-by-step explanation:
1425 ------ 45600
1820 ------ x
1425*x = 1425x
45600*1820 = 82992000
1425x = 82992000
82992000/1425 = x
58240 = x
Keep on dividing them both by 2
