Linear Transformation

In summary: The correct matrix is as others have indicated: \begin{bmatrix}0 & 0 & 1\\ -1 & 0 & 0\\ 1 & 0 & 1\end{bmatrix}In summary, the standard matrix representation for T1 is found by applying T1 to each basis vector and using the resulting columns as the matrix. The resulting matrix is then multiplied by the given vector [T1(1,-3,0)]E to find the final output, which in this case is [0,-1,0]. The correct matrix representation is: \begin{bmatrix}0 & 0 & 1\\ -1 & 0 & 0\\ 1 &
  • #1
DanielFaraday
87
0

Homework Statement



Write the standard matrix representation for T1 and use it to find [T1(1,-3,0)]E.

Homework Equations



[tex]
T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right)
[/tex]

The Attempt at a Solution



I just wanted to check to see if I am doing this right. Thanks in advance!

[tex]
A=\left(
\begin{array}{ccc}
0 & 0 & 1 \\
-1 & 0 & 0 \\
0 & 0 & 1
\end{array}
\right)\
[/tex]

[tex]
\left[T_1(1,-3,0)\right]_E=A\left(
\begin{array}{c}
1 \\
-3 \\
0
\end{array}
\right)=\left(
\begin{array}{ccc}
0 & 0 & 1 \\
-1 & 0 & 0 \\
0 & 0 & 1
\end{array}
\right).\left(
\begin{array}{c}
1 \\
-3 \\
0
\end{array}
\right)=\left(
\begin{array}{c}
0 \\
-1 \\
0
\end{array}
\right)
[/tex]
 
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  • #2
Your A is transposed from what it should be.
 
  • #3
No, Officeshredder,
Applying T1 to each basis vector in turn gives the columns.

T1(1, 0, 0)= (0, -1, 0)
T1(0, 1, 0)= (0, 0, 0)
T1(0, 0, 1)= (1, 0, 1)

So T1 is represented by
[tex]\begin{bmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 1 & 0 & 1 \end{bmatrix}[/tex]
exactly what DanielFaraday has.

And, of course, T1(1,-3,0)= (0,-1,0) as said.
 
  • #4
Thank you both for your input!
 
  • #5
Office shredder may be using a different convention than you and I:

[tex]T_1(1,-3,0)= \begin{bmatrix}1 & -3 & 0\end{bmatrix}\begin{bmatrix}1 & -1 & 0\\ 0 & 0 & 0 \\ 1 & 0 & 1\end{bmatrix}= \begin{bmatrix}0 \\ -1 \\ 0\end{bmatrix}[/tex]
 
  • #6
Yes, these things often depend on the textbook. Thanks.
 
  • #7
No, sorry, that was just a brain fart on my part
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector from one vector space to another vector space in a way that preserves the basic properties of vector addition and scalar multiplication.

2. What are the key properties of a linear transformation?

The key properties of a linear transformation are that it is additive, homogeneous, and preserves the zero vector. This means that the transformation of the sum of two vectors is equal to the sum of their individual transformations, the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformation of the original vector, and the transformation of the zero vector is equal to the zero vector.

3. How is a linear transformation represented?

A linear transformation can be represented by a matrix. The columns of the matrix represent the images of the standard basis vectors of the input vector space, and the transformation of any vector can be found by multiplying the matrix by the vector.

4. What is the role of a linear transformation in linear algebra?

Linear transformations are fundamental to the study of linear algebra as they allow us to understand how geometric objects, such as vectors and matrices, behave when transformed. They also help us solve systems of linear equations, find eigenvalues and eigenvectors, and perform other important operations in linear algebra.

5. How are linear transformations used in real life?

Linear transformations are used in many real-life applications, such as computer graphics, image processing, and data analysis. They are also used in physics and engineering to model and solve problems involving linear systems. Additionally, linear transformations are used in economics and finance to analyze linear relationships and make predictions.

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