Linear Transformation Question

seansrk

Question about linear transformations if you have a matrix such as

| 5 6 9 |
| 5 0 3 |
| 9 -3 -7 |

Can it be a matrix transformation? Or does it have to follow the identity matrix?
Can be a transformation and the "y" transformation being just makes the it flat on the y axis? or does it have to be a form of the identity matrix?

Or am I totally misunderstanding this?

Alchemista

A matrix is a linear transformation expressed with respect to a basis for the source space and the target space.

Given a linear transformation [itex]T:\mathbb{F}^n \to \mathbb{F}^m[/itex], the corresponding matrix written with respect to a basis [itex]\alpha[/itex] for the source space and a basis [itex]\beta[/itex] for the target space is as follows:

[itex]
\left[
\begin{array}{cccc}
[T(\alpha_1)]_\beta & [T(\alpha_2)]_\beta & ... & [T(\alpha_n)]_\beta
\end{array}
\right]
[/itex]

The theory behind this is as follows. Since any vector in a given vector space can be expressed as a linear combination of a set of basis vectors for that vector space, we need only transform an arbitrary basis to capture the transformation.

Given some vector [itex]v \in \mathbb{F}^n[/itex] and a basis [itex]\alpha[/itex] we can write [itex]v = a_1\alpha_1 + a_2\alpha_2 + ... + a_n\alpha_n[/itex]. Then [itex]v[/itex] transformed is as follows

[itex]
\begin{eqnarray*}
T(v) &=& T(a_1\alpha_1 + a_2\alpha_2 + ... + a_n\alpha_n) \\
&=& T(a_1\alpha_1) + T(a_2\alpha_2) + ... + T(a_n\alpha_n) \\
&=& a_1T(\alpha_1) + a_2T(\alpha_2) + ... + a_nT(\alpha_n)
\end{eqnarray*}
[/itex]

Fredrik

I don't understand your questions. I don't know what you mean by "follow the identity matrix" or "a form of the identity matrix". Also, how do you define "matrix transformation" if you don't mean "a function defined by a matrix"?

HallsofIvy

Any m by n matrix is a linear transformation from [itex]R^m[/itex] to [itex]R^n[/itex].

What you have given is a perfectly good linear transformation- although the way you have written it, with the "straight" vertical sides, makes it look more like a determinant than a matrix!

The matrix you give represents the linear transformation that maps a vector, [itex]a\vec{i}+ b\vec{j}+ c\vec{k}[itex] into [itex]a(5\vec{i}+ 5\vec{j}+ 9\vec{k})+ b(6\vec{i}- 3\vec{k})+ c(9\vec{i}+ 3\vec{j}- 7\vec{kl})[/itex][itex]= (5a+ 6b+ 9c)\vec{i}+ (5a+ 3b)\vec{i}+ (9a- 3b- 7c)\vec{k}[/itex].

I wonder if you aren't confusing "matrix", in general, with "invertible matrix".