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I mean a basis of vectors for the 2-dimensional vector space R^2Basis for what?
Yes, you are right. My amended version is to get the change-of-basis matrix from basis A=(1,0),(4,7) to basis B=(0,2),(2,1)(4,7) is not a basis for $\Bbb R^2$, which having dimension 2, needs a basis of 2 vectors.
Say $(b_1,b_2)$ is an (ordered) basis for $\mathbb{R}^2$ and $(b_1',b_2')$ is another (ordered) basis for $\mathbb{R}^2$. We can express $b_1' = Ab_1 + Bb_2$ and $b_2' = Cb_1 + Db_2$. Then the change of basis matrix from $(b_1,b_2)$ to $(b_1',b_2')$ is given by,Hi
How can I get the change-of-basis matrix from basis (4,7) to basis (2,1)
Also, how do I use it once I get it.
Hi dingo!Yes, you are right. My amended version is to get the change-of-basis matrix from basis A=(1,0),(4,7) to basis B=(0,2),(2,1)
I think one has to solve for $(x_A,y_A)$ to find the change-of-basis matrix:Solve for $\begin{pmatrix}{x_B\\y_B}\end{pmatrix}$ to find the change-of-basis matrix.
As I interpret it, the change-of-basis matrix from basis A to basis B would convert a vector with respect to basis A to a vector with respect to basis B.I think one has to solve for $(x_A,y_A)$ to find the change-of-basis matrix:
\[
\begin{pmatrix}{x_A\\y_A}\end{pmatrix}=C \begin{pmatrix}{x_B\\y_B}\end{pmatrix}
\]
where $C$ is the required matrix.
(To the OP: this is a theorem, not the definition of a change-of-basis matrix.)
There is a type mismatch here. In order for the notation $[v]_C$ to make sense, $v$ has to be a pair of coordinates rather than a vector. But $T$ acts on vectors, not coordinates, so then $T(v)$ does not make sense. I assume that, contrary to the previous convention, $[v]_C$ denotes the coordinates of $v$ in the basis $C$.So, given a basis $B = \{v_1,v_2\}$ of a two-dimensional vector space $V$ over a field $F$, I will write:
$[c_1,c_2]_B$ as shorthand for the formal linear combination:
$c_1v_1 + c_2v_2$.
It is important to realize that a matrix $A$ DOES NOT REPRESENT a linear transformation, per se. It rather represents a linear transformation GIVEN bases for the domain space, and co-domain space. If our linear transformation is $T$, we might write:
$A = [T]_B^C$
to denote that:
$[T]_B^C[v]_C = [T(v)]_B$
Yes, I am "abusing the notation" somewhat to write:There is a type mismatch here. In order for the notation $[v]_C$ to make sense, $v$ has to be a pair of coordinates rather than a vector. But $T$ acts on vectors, not coordinates, so then $T(v)$ does not make sense. I assume that, contrary to the previous convention, $[v]_C$ denotes the coordinates of $v$ in the basis $C$.