91Ó°ÊÓ

Matrix-vector multiplication is a fundamental operation in linear algebra that involves a matrix and a vector. Given a matrix \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and a vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \), the multiplication \( A\mathbf{x} \) results in another vector. This operation can be seen as a series of dot products between rows of the matrix and the vector itself.

To illustrate, consider the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \). The multiplication proceeds as follows:

• Compute the dot product of the first row of \( A \) with \( \mathbf{x} \): \( y_1 = 1 \cdot x_1 + 2 \cdot x_2 = x_1 + 2x_2 \).
• Compute the dot product of the second row: \( y_2 = 3 \cdot x_1 + 4 \cdot x_2 = 3x_1 + 4x_2 \).

This results in the vector \( \begin{bmatrix} y_1 \ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + 2x_2 \ 3x_1 + 4x_2 \end{bmatrix} \), showcasing how matrix-vector multiplication transforms the input vector.

A linear map, in the context of linear algebra, is a function that models the transformation of vectors from one vector space to another, when represented by a matrix. The map preserves vector addition and scalar multiplication properties. For instance, if \( A: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) represents a linear map, as seen with matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), each input vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) generates outputs through component functions.

These component functions are derived from matrix-vector multiplication, and for \( A \), they are:

• \( y_1 = x_1 + 2x_2 \)
• \( y_2 = 3x_1 + 4x_2 \)

Thus, the transformation \( A \mathbf{x} = \begin{bmatrix} x_1 + 2x_2 \ 3x_1 + 4x_2 \end{bmatrix} \) illustrates the action of the linear map. Properties include preserving linear combinations, ensuring that transformations maintain structure, and mapping straight lines to straight lines or points in a geometrical sense.

The continuity of linear transformations is a key concept in both real analysis and linear algebra. For a linear map \( A: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), continuity means that small changes in the input vector \( \mathbf{x} \) result in small changes in the output \( A(\mathbf{x}) \).

To establish this, consider a sequence of vectors \( \mathbf{x}_n \rightarrow \mathbf{x} \) in \( \mathbb{R}^2 \). The corresponding outputs \( A(\mathbf{x}_n) \) must converge to \( A(\mathbf{x}) \). For the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), this translates mathematically to:

• If \( \mathbf{x}_n = \begin{bmatrix} x_{1,n} \ x_{2,n} \end{bmatrix} \rightarrow \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \), then \( A(\mathbf{x}_n) = \begin{bmatrix} x_{1,n} + 2x_{2,n} \ 3x_{1,n} + 4x_{2,n}\end{bmatrix} \rightarrow \begin{bmatrix} x_1 + 2x_2 \ 3x_1 + 4x_2 \end{bmatrix} = A(\mathbf{x}) \).

This compliance with the continuity criterion is ensured because all operations involved (addition and multiplication) are continuous. Hence, linear maps described by matrices with real coefficients are inherently continuous.