91Ó°ÊÓ

In linear algebra, when dealing with linear transformations, the concept of a 'Standard Matrix' is central. Linear transformations can often be expressed as matrix equations. This involves transforming input vectors using a set matrix of coefficients. For a transformation \( T(x, y, z) = (3x - 2y + z, 2x - 3y, y - 4z) \), each component of the output is a linear combination of the input variables \( x, y, \) and \( z \).

To derive the 'Standard Matrix', we arrange these coefficients into a matrix. Consider each equation separately:

• For the first term: \( 3x - 2y + z \)
• For the second term: \( 2x - 3y \)
• For the third term: \( y - 4z \)

These terms allow us to form a matrix \( A \), where each row corresponds to an equation:

• First row: coefficients of \( x, y, \) and \( z \) are \([3, -2, 1]\)
• Second row: \([2, -3, 0]\)
• Third row: \([0, 1, -4]\)

Thus, the standard matrix \( A \) representing the transformation \( T \) is \\[\begin{bmatrix} 3 & -2 & 1 \ 2 & -3 & 0 \ 0 & 1 & -4 \end{bmatrix}\]. This matrix captures the essence of how the linear transformation alters input vectors.

Once we have the standard matrix, we can use it to transform vectors through 'Matrix-Vector Multiplication'. This operation allows us to find what happens to a vector when it undergoes a given linear transformation.

Assuming the vector \( \mathbf{v} = (2, -1, -1) \), and the standard matrix \( A \) as:\[\begin{bmatrix} 3 & -2 & 1 \ 2 & -3 & 0 \ 0 & 1 & -4 \end{bmatrix}\],we'll do the following multiplication:

• Multiply each element of the vector with the corresponding elements in each row of the matrix.
• The first element of the result is \( 3 \times 2 + (-2) \times (-1) + 1 \times (-1) \).
• Continue for the other rows to find the full resulting vector.

Through matrix-vector multiplication, each component of the vector undergoes transformation according to the rules defined by matrix \( A \). This provides us the image of the vector under transformation \( T \). This very abstract concept illustrates how transformations can be cleanly handled through matrix operations.

The 'Image of a Vector' is the resultant vector after applying the linear transformation via its standard matrix. When working with a specific vector, like \( \mathbf{v} = (2, -1, -1) \), and the given transformation matrix \( A \), we find its 'image' by performing the matrix-vector multiplication explained earlier.

To calculate the image of \( \mathbf{v} \), simply follow these steps:

• Use the standard matrix \( A \).
• Multiply \( A \) by \( \mathbf{v} \).
• The result is the transformed vector.

For our specific case with: \[\begin{bmatrix} 3 & -2 & 1 \ 2 & -3 & 0 \ 0 & 1 & -4 \end{bmatrix}\begin{bmatrix} 2 \ -1 \ -1 \end{bmatrix}\] the computed image is a new vector. Understanding this concept is paramount, as it tells us how different inputs are transformed and what outputs, or images, they map onto. This operation is a crucial tool in visualizing and computing transformations in linear algebra.