notation

A  =  {{1, -1}, {1, 1}, {2, 0}}

MatrixForm[A]

x  =  {1, 2}
MatrixForm[x]
y  =  {1, 2, 3}
MatrixForm[y]

Transpose[x]

xt  =  {{1}, {2}}
MatrixForm[xt]

A  .  y

y  .  A

solving linear systems

B  =  {{1, 1}, {1, -1}}
y  =  {-1, 1}
MatrixForm[B]
MatrixForm[Inverse[B]]

MatrixForm[Inverse[B]  .  B]

B  .  B

.5 B  .  B

B B

MatrixForm[B]  . MatrixForm[B]

x  =  Inverse[B] .  y

B  .  x
y

extreme numerical example

m = 
        N[Table[1/(i + j - 1), {i, 6}, {j, 6}], 
          20];
MatrixForm[m]
MatrixForm[m  .  Inverse[m]]
MatrixForm[Chop[m . Inverse[m]]]

rectangular linear systems:  what do they mean?

A  =  {{1, -1}, {1, 1}, {2, 2}};
y  =  {0, 1, 2};
MatrixForm[A]
MatrixForm[y]

PI  =   PseudoInverse[A];
MatrixForm[PI]

PI  .  y

A  .  %

this solution solves all three equations even though there are only two unknowns .   But what about the next one?

A  =  {{1, -1}, {1, 1}, {2, 2}};
y  =  {0, 1, 1};
MatrixForm[A]
MatrixForm[y]

PI  =   PseudoInverse[A];
MatrixForm[PI]
PI . y

A .  %