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Firstly, my apologies to Deveno in the event that he has already answered these questions in a previous post ...
Now ...
Suppose we have a linear transformation $$T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2$$ , say ...
Suppose also that $$\mathbb{R}^3$$ has basis $$B$$ and $$\mathbb{R}^2$$ has basis $$B'$$, neither of which is the standard basis ...
Suppose further that $$T(x, y, z) = ( x + 2y - z , 3x + 5z )$$ ... ...
... ... ...
Then (if I am right) we write the matrix $$A$$, of the transformation as follows:
$$A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 0 & 5 \end{bmatrix}$$... BUT ... questions ...Question 1
Is the expression for the linear transformation
$$T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2$$
an expression in terms of the transformation of $$v = (x, y, z)$$ into $$w = T(v)$$ in terms of the bases $$B$$ and $$B'$$ ... ...
That is, when we input some vector $$v = ( 2, 1, -3 )$$ , say ... ... is that vector to be read as being in terms of the basis $$B$$ or in terms of the standard basis ... ...
... ... and is the output vector from applying T, namely
$$T(v) = ( 2, 1, 3 ) = ( x + 2y - z , 3x + 5z ) = ( 2 + 2(1) - (-3) , 3(2) + 5(-3) ) = ( 7, -9 )$$
in terms of the basis $$B'$$ or in terms of the standard basis?[By the way, I think it is, by convention, that linear transformations from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$ are expressed as if they are from a standard basis to a standard basis ... but why they are not taken to be in the declared bases $$B$$ and $$B'$$, I am not sure ... ... ]
Question 2
Does the matrix of the transformation
$$A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 0 & 5 \end{bmatrix}$$
represent the transformation from $$[v]_B$$ to $$[T(v)]_{B'}$$
or
does it represent the transformation from $$[v]_{S_1}$$ to $$[T(v)]_{S_2}$$
where $$S_1$$ is the standard basis for $$\mathbb{R}^3$$
and $$S_2$$ is the standard basis for $$\mathbb{R}^2$$
Hope someone can help ...
Peter
Now ...
Suppose we have a linear transformation $$T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2$$ , say ...
Suppose also that $$\mathbb{R}^3$$ has basis $$B$$ and $$\mathbb{R}^2$$ has basis $$B'$$, neither of which is the standard basis ...
Suppose further that $$T(x, y, z) = ( x + 2y - z , 3x + 5z )$$ ... ...
... ... ...
Then (if I am right) we write the matrix $$A$$, of the transformation as follows:
$$A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 0 & 5 \end{bmatrix}$$... BUT ... questions ...Question 1
Is the expression for the linear transformation
$$T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2$$
an expression in terms of the transformation of $$v = (x, y, z)$$ into $$w = T(v)$$ in terms of the bases $$B$$ and $$B'$$ ... ...
That is, when we input some vector $$v = ( 2, 1, -3 )$$ , say ... ... is that vector to be read as being in terms of the basis $$B$$ or in terms of the standard basis ... ...
... ... and is the output vector from applying T, namely
$$T(v) = ( 2, 1, 3 ) = ( x + 2y - z , 3x + 5z ) = ( 2 + 2(1) - (-3) , 3(2) + 5(-3) ) = ( 7, -9 )$$
in terms of the basis $$B'$$ or in terms of the standard basis?[By the way, I think it is, by convention, that linear transformations from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$ are expressed as if they are from a standard basis to a standard basis ... but why they are not taken to be in the declared bases $$B$$ and $$B'$$, I am not sure ... ... ]
Question 2
Does the matrix of the transformation
$$A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 0 & 5 \end{bmatrix}$$
represent the transformation from $$[v]_B$$ to $$[T(v)]_{B'}$$
or
does it represent the transformation from $$[v]_{S_1}$$ to $$[T(v)]_{S_2}$$
where $$S_1$$ is the standard basis for $$\mathbb{R}^3$$
and $$S_2$$ is the standard basis for $$\mathbb{R}^2$$
Hope someone can help ...
Peter