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What is the matrix in the canonical basis of a linear transformation?

An example:

Let $s:\Bbb R^2\to\Bbb R^2$ a linear transformation whose matrix in the canonical basis is $[t]_c=\begin{bmatrix}1&5\\2&-2\end{bmatrix}$

What it means?

And if I have to find the matrix in the canonical basis?

Thanks!

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This means that your basis in $c$ is $[t]_c$. Now you have to find $\scr{B}$ such that $[t]_{\scr{B}} = [t]_c$.

This means you have to diagonalize your matrix. You may or may not have to get it in Jordan normal form, depending on if its eigenvectors are L.I. hope this helps

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  • $\begingroup$ This is nonsense. Did you actually read the question? There is no $\mathscr B$ at all in it. $\endgroup$ – Marc van Leeuwen Dec 7 '13 at 13:11
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The canonical basis only exists for vector spaces of the form $\Bbb R^n$ (or $K^n$ if the field is $K$), in other words spaces whose elements are lists of $n$ scalars (although the existence of other kinds of finite dimensional vector spaces is largely ignored in many linear algebra courses, most actual problems in which linear algebra is invoked start out with such a different kind of vector space, for instance a space of functions). Now the canonical basis is the one whose vectors are the columns of the $n\times n$ identity matrix. In the case of $\Bbb R^2$, it is $\binom10,\binom01$. Saying "a linear transformation whose matrix in the canonical basis is $A$" means interpreting $A$ as a linear map in the most obvious way: the linear map that sends $v\mapsto A\cdot v$.

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A linear transformation (LT) is characterized entirely by its action on a basis, since all other vectors can be made out of a linear combination of that basis. Thus, if you specify LT{basis vectors}, you've essentially told me how the entire LT works.

This information can be captured in a matrix. Thus, the matrix represents the LT. However, as mentioned above, what the matrix describes is where the basis vectors get mapped to. Since there are multiple possible bases, the matrix representation is not unique. The same LT described with respect to different bases gets captured as a different matrix.

In your case, the "matrix in the canonical basis" means that the LT is being captured as a matrix with respect to the canonical basis (i.e. the standard basis).

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