In the right figure you have the first
example of a not canonical base ;
it is formed by two vectors not null neither parallel
and
(they are also unit and perpendicular, like
and
, but we
will talk about that in the next
activity).
The components
of another vector
in this
base are the two scalars x and y that they let
write
like
a linear combination of
and
:
=
x
+ y
.
Like this, in the case of the figure, vector
can express in in two ways:
= 1,3
+ 4![](gif/vec_j.gif)
=
3.3
+ 2.6![](gif/vec_n.gif)
therefore, while the
components of
in the canonical base
,
are
(1.3 , 4), components of
in the base not canonical
,
are (3.3 , 2.6).
To pass to
the expression of
in the base
,
to the expression of
in the base
,
we have to know how to write vectors
and
like a
linear combination of the vectors
and
(or how to
express
and
like a linear combination of
and
);
they are called formulas of changing of base.
In the homework we will use them.