18.06.05: ‘Systems of linear
equations as matrices’
Lecturer: Barwick
Friday 12 February 2016
18.06.05: ‘Systems of linear equations as matrices’
Suppose
𝑎11 𝑎12
𝑎
𝑎22
𝐴 = ( 21
⋮ ⋮
𝑎𝑚1 𝑎𝑚2
⋯ 𝑎1𝑛
⋯ 𝑎2𝑛
)
⋱ ⋮
⋯ 𝑎𝑚𝑛
an 𝑚 × 𝑛 matrix, and
𝑣1
𝑣
( 2 ) ∈ R𝑚
⋮
𝑣𝑛
a vector.
18.06.05: ‘Systems of linear equations as matrices’
Last time, we learned that the following are logically equivalent:
1. 𝑣 ⃗ = 𝐴𝑥⃗ for some vector 𝑥⃗ ∈ R𝑛 ;
2. the system of linear equations
𝑣1
=
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 ;
𝑣2
=
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 ;
⋮
𝑣𝑚
=
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 ;
has at least one solution;
3. 𝑣 ⃗ lies in the span of the column vectors 𝐴1⃗ , 𝐴2⃗ , … , 𝐴𝑛⃗ .
18.06.05: ‘Systems of linear equations as matrices’
Can this system of equations be solved?
1 = 2𝑥 + 4𝑦;
1 = 6𝑥 + 8𝑧;
1 = 10𝑦 + 12𝑧.
18.06.05: ‘Systems of linear equations as matrices’
How about this one?
3 = 2𝑥 + 5𝑦;
−7 = 3𝑥 + 3𝑦;
3 = 5𝑥 + 2𝑦.
18.06.05: ‘Systems of linear equations as matrices’
This one?
3 = 2𝑥 + 5𝑦;
0 = 3𝑥 + 3𝑦;
−3 = 5𝑥 + 2𝑦.
18.06.05: ‘Systems of linear equations as matrices’
The following are logically equivalent:
1. there are infinitely many vectors 𝑥⃗ such that 𝐴𝑥⃗ = 0;
2. there is a nonzero vector 𝑥⃗ ∈ R𝑛 that is orthogonal to all of the row
vectors 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 ;
3. the row vectors 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 do not span all of R𝑛 .
18.06.05: ‘Systems of linear equations as matrices’
Why are those last two the same?
On the one hand, if there’s a vector 𝑥⃗ ∈ R𝑛 that is orthogonal to all of the row
vectors 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 , then certainly the row vectors 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 do not
span all of R𝑛 .
But what about the other direction? If there’s a vector 𝑥⃗ not in the span of
𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 , then it doesn’t have to be orthogonal to each of 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚
…
18.06.05: ‘Systems of linear equations as matrices’
However, there’s a procedure (called the Gram–Schmidt process) that produces
from 𝑥⃗ a new vector 𝑦⃗ that is orthogonal to each of the vectors 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 .
We will return to this later in the course!
18.06.05: ‘Systems of linear equations as matrices’
How many solutions have we here?
0 = 2𝑥 + 4𝑦;
0 = 4𝑦 + 6𝑧.
18.06.05: ‘Systems of linear equations as matrices’
How many here?
0 = 𝑥 + 𝑦 − 𝑧;
0 = 𝑥 − 𝑦 + 𝑧;
0 = −𝑥 + 𝑦 + 𝑧.
18.06.05: ‘Systems of linear equations as matrices’
Here’s a fun exercise: in R4 , can we find four vectors 𝑣1⃗ , 𝑣2⃗ , 𝑣3⃗ , 𝑣4⃗ such that the
angle between any pair of them is 𝜋/4?
18.06.05: ‘Systems of linear equations as matrices’
Here’s one that might come as a surprise: suppose 𝐴 an 𝑛 × 𝑛 matrix. Then the
following are logically equivalent:
⃗ are linearly independent;
1. the column vectors 𝐴1⃗ , 𝐴2⃗ , … , 𝐴𝑚
2. the row vectors 𝐴⃗ 1 , 𝐴⃗ 2 , … , 𝐴⃗ 𝑚 are linearly independent;
18.06.05: ‘Systems of linear equations as matrices’
3. the system of linear equations
0
=
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 ;
0
=
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 ;
⋮
0
=
has exactly one solution.
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 ;
18.06.05: ‘Systems of linear equations as matrices’
Here’s a system of 512 linear equations in 512 variables 𝑥1 , 𝑥2 , … , 𝑥512 :
513
=
𝑥2 + 𝑥3 + ⋯ + 𝑥511 + 𝑥512 ;
514
=
𝑥1 + 𝑥3 + ⋯ + 𝑥511 + 𝑥512 ;
⋮
1023
=
𝑥1 + 𝑥2 + ⋯ + 𝑥510 + 𝑥512 ;
1024
=
𝑥1 + 𝑥2 + ⋯ + 𝑥510 + 𝑥511 .
How many solutions does it have?