Page 375 - 35Linear Algebra

G.2 Systems of Linear Equations 375

Hint for Review Question 5

The first part for Review Question 5 is simple--just write out the associated
linear system and you will find the equation 0 = 6 which is inconsistent.
Therefore we learn that we must avoid a row of zeros preceding a non-vanishing
entry after the vertical bar.
Turning to the system of equations, we first write out the augmented matrix
and then perform two row operations
 
1 −3 0 6
 1 0 3 −3 
2 k 3 − k 1
 
1 −3 0 6
R 2 −R 1 ;R 3 −2R 1  0
∼ 3 3 −9  .
0 k + 6 3 − k −11
Next we would like to subtract some amount of R 2 from R 3 to achieve a zero in
the third entry of the second column. But if
3
k + 6 = 3 − k ⇒ k = − ,
2
this would produce zeros in the third row before the vertical line. You should
also check that this does not make the whole third line zero. You now have
enough information to write a complete solution.

Planes

Here we want to describe the mathematics of planes in space. The video is
summarised by the following picture:

2
A plane is often called R because it is spanned by two coordinates, and space
3
is called R and has three coordinates, usually called (x, y, z). The equation
for a plane is
ax + by + cz = d .

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