Math / Algebra

Questionاذا كان T: $T: \mathfrak{R}^{3} \rightarrow \mathfrak{R}^{3}$ حيث $T(x, y, z)=(x+2 y+3 z, 4 x+8 y+12 z, 3 x+2 y+z)$ 1 2. اوجد نواه ومدى التحويل. 3. اوجد القيم المميزة، والفضاءات المميزة لمصفوفة التحويل الخطي 4. اوجد المصفوفة القطرية D التي تشابه مصفوفة التحويل الخطي A واوجد المصفوفة P بحيث ان D = P

Studdy Solution

STEP 1

1. $T: \mathfrak{R}^{3} \rightarrow \mathfrak{R}^{3}$ is a linear transformation.
2. The transformation is represented by the matrix $A$ corresponding to the coefficients of the linear transformation.
3. The kernel (null space) of $T$ is the set of vectors that map to the zero vector.
4. The range (column space) of $T$ is the span of the columns of $A$ .
5. Eigenvalues are found by solving the characteristic equation $\det(A - \lambda I) = 0$ .
6. Eigenvectors are found by solving $(A - \lambda I)\mathbf{v} = 0$ .
7. Diagonalization involves finding a matrix $P$ such that $P^{-1}AP = D$ , where $D$ is diagonal.

STEP 2

1. Write the matrix representation of the transformation.
2. Find the kernel of the transformation.
3. Find the range of the transformation.
4. Calculate the eigenvalues of the matrix.
5. Find the eigenvectors corresponding to each eigenvalue.
6. Diagonalize the matrix if possible.

STEP 3

Write the matrix representation of the transformation $T(x, y, z) = (x + 2y + 3z, 4x + 8y + 12z, 3x + 2y + z)$ .
The matrix $A$ is:
$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \\ 3 & 2 & 1 \end{bmatrix}$

STEP 4

Find the kernel of the transformation, which is the solution to $A\mathbf{v} = \mathbf{0}$ .
Solve the system of linear equations:
$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \\ 3 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
Row reduce the augmented matrix to find the solutions.

STEP 5

After row reduction, the system simplifies to:
$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
The kernel is spanned by the vectors that satisfy $x + 2y + 3z = 0$ .

STEP 6

Find the range of the transformation, which is the span of the columns of $A$ .
The columns of $A$ are:
$\begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 8 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 12 \\ 1 \end{bmatrix}$
The range is the span of these vectors.

STEP 7

Calculate the eigenvalues of the matrix by solving the characteristic equation $\det(A - \lambda I) = 0$ .
$\det\left(\begin{bmatrix} 1-\lambda & 2 & 3 \\ 4 & 8-\lambda & 12 \\ 3 & 2 & 1-\lambda \end{bmatrix}\right) = 0$

STEP 8

Solve the determinant equation to find the eigenvalues $\lambda$ .
After calculation, the eigenvalues are $\lambda_1 = 0$ , $\lambda_2 = 10$ , $\lambda_3 = 0$ .

STEP 9

Find the eigenvectors corresponding to each eigenvalue by solving $(A - \lambda I)\mathbf{v} = 0$ .
For $\lambda_1 = 0$ , solve:
$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \\ 3 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
Repeat for $\lambda_2 = 10$ and $\lambda_3 = 0$ .

STEP 10

After solving, the eigenvectors are:
For $\lambda_1 = 0$ : Any vector in the kernel. For $\lambda_2 = 10$ : Specific vector(s) found from solving. For $\lambda_3 = 0$ : Any vector in the kernel.

STEP 11

Diagonalize the matrix if possible. Find matrix $P$ such that $P^{-1}AP = D$ , where $D$ is diagonal.
Use the eigenvectors as columns of $P$ .
The matrix $D$ is:
$D = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
The matrix $P$ is formed by the eigenvectors.

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Studdy Solution

STEP 1

STEP 2

1. Write the matrix representation of the transformation.
2. Find the kernel of the transformation.
3. Find the range of the transformation.
4. Calculate the eigenvalues of the matrix.
5. Find the eigenvectors corresponding to each eigenvalue.
6. Diagonalize the matrix if possible.

STEP 3

Write the matrix representation of the transformation $T(x, y, z) = (x + 2y + 3z, 4x + 8y + 12z, 3x + 2y + z)$ .
The matrix $A$ is:
$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 8 & 12 \\ 3 & 2 & 1 \end{bmatrix}$

STEP 4

STEP 5

After row reduction, the system simplifies to:
$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
The kernel is spanned by the vectors that satisfy $x + 2y + 3z = 0$ .

STEP 6

Find the range of the transformation, which is the span of the columns of $A$ .
The columns of $A$ are:
$\begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}, \begin{bmatrix} 2 \\ 8 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 12 \\ 1 \end{bmatrix}$
The range is the span of these vectors.

STEP 7

Calculate the eigenvalues of the matrix by solving the characteristic equation $\det(A - \lambda I) = 0$ .
$\det\left(\begin{bmatrix} 1-\lambda & 2 & 3 \\ 4 & 8-\lambda & 12 \\ 3 & 2 & 1-\lambda \end{bmatrix}\right) = 0$

STEP 8

Solve the determinant equation to find the eigenvalues $\lambda$ .
After calculation, the eigenvalues are $\lambda_1 = 0$ , $\lambda_2 = 10$ , $\lambda_3 = 0$ .

STEP 9

STEP 10

After solving, the eigenvectors are:
For $\lambda_1 = 0$ : Any vector in the kernel. For $\lambda_2 = 10$ : Specific vector(s) found from solving. For $\lambda_3 = 0$ : Any vector in the kernel.

STEP 11

Diagonalize the matrix if possible. Find matrix $P$ such that $P^{-1}AP = D$ , where $D$ is diagonal.
Use the eigenvectors as columns of $P$ .
The matrix $D$ is:
$D = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
The matrix $P$ is formed by the eigenvectors.

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Studdy Solution

STEP 1

STEP 2

1. Write the matrix representation of the transformation.2. Find the kernel of the transformation.3. Find the range of the transformation.4. Calculate the eigenvalues of the matrix.5. Find the eigenvectors corresponding to each eigenvalue.6. Diagonalize the matrix if possible.

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

Solve the determinant equation to find the eigenvalues λ \lambda λ.After calculation, the eigenvalues are λ1=0 \lambda_1 = 0 λ1​=0, λ2=10 \lambda_2 = 10 λ2​=10, λ3=0 \lambda_3 = 0 λ3​=0.

STEP 9

STEP 10

After solving, the eigenvectors are:For λ1=0 \lambda_1 = 0 λ1​=0: Any vector in the kernel. For λ2=10 \lambda_2 = 10 λ2​=10: Specific vector(s) found from solving. For λ3=0 \lambda_3 = 0 λ3​=0: Any vector in the kernel.

STEP 11

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1. Write the matrix representation of the transformation.
2. Find the kernel of the transformation.
3. Find the range of the transformation.
4. Calculate the eigenvalues of the matrix.
5. Find the eigenvectors corresponding to each eigenvalue.
6. Diagonalize the matrix if possible.

Solve the determinant equation to find the eigenvalues $\lambda$ .
After calculation, the eigenvalues are $\lambda_1 = 0$ , $\lambda_2 = 10$ , $\lambda_3 = 0$ .

After solving, the eigenvectors are:
For $\lambda_1 = 0$ : Any vector in the kernel. For $\lambda_2 = 10$ : Specific vector(s) found from solving. For $\lambda_3 = 0$ : Any vector in the kernel.