Can This Matrix Represent a Linear Transformation?

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

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

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 $v \in \mathbb{F}^n$ and a basis $\alpha$ we can write $v = a_1\alpha_1 + a_2\alpha_2 + ... + a_n\alpha_n$. Then $v$ transformed is as follows

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

 

[Fredrik]

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?

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 $R^m$ to $R^n$.

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 $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})$$= (5a+ 6b+ 9c)\vec{i}+ (5a+ 3b)\vec{i}+ (9a- 3b- 7c)\vec{k}$.

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

 

[blue_raver22]

A linear transformation is a mathematical function that maps one vector space onto another vector space while preserving the linear structure of the original space. In the context of matrices, a linear transformation is represented by a matrix multiplication.

In the given matrix, each entry represents a transformation of the corresponding vector. For example, the first column represents the transformation of the x-axis, the second column represents the transformation of the y-axis, and the third column represents the transformation of the z-axis.

To answer the question, the matrix does not necessarily have to follow the identity matrix. The matrix can represent any linear transformation as long as it preserves the linear structure of the original space. The identity matrix is just a specific type of linear transformation that preserves the original space.

In this case, the matrix can be a transformation and the "y" transformation can flatten the vector on the y-axis, but it does not have to be in the form of the identity matrix. It is important to note that a linear transformation does not necessarily have to change the shape of the vector. It can also just change the direction or magnitude.

Overall, it seems like there may be some misunderstanding about linear transformations. It is important to understand that a linear transformation is a mathematical function that can be represented by a matrix, but it does not have to follow the identity matrix.