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Understanding the basics of coordinate geometry is key to solving problems related to the area of shapes on a coordinate plane. The foundation of coordinate geometry lies in the Cartesian coordinate system where any point on a plane is described by an x (horizontal) and y (vertical) coordinate.

When dealing with triangles, the vertices are essentially points on this plane. In our exercise, we have the vertices at \( (0,0),(2,0),(0,3) \), which represent the corners of the triangle. Visualizing this, we would see that the triangle lies in the first quadrant of the Cartesian plane, with the right angle at the origin. By plotting these points, students can gain a better understanding of the triangle's shape and orientation, which is the start of finding its area using the vertices.

The area of a triangle is a measurement of the space enclosed within its three sides. To calculate this area using coordinates, the triangle area formula via determinants comes into play. This formula is derived from the concept of the absolute value of a determinant of a matrix in linear algebra.

The formula looks like this: \[ Area = 0.5 * |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| \]. By plugging the coordinates of the triangle's vertices into this equation, one can find the area without needing to calculate base and height explicitly. This formula can be particularly helpful when triangles are not right-angled or when they do not sit on a horizontal base, making traditional methods less straightforward.

Linear algebra is a branch of mathematics primarily concerned with vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also applicable to more complicated structures. In terms of the triangle area problem, linear algebra provides tools such as matrices and determinants that play a crucial role.

In our step-by-step solution to find the area of a triangle with given vertices, linear algebra surfaces in the form of the determinant. This form is a high-level application of linear algebra where we calculate a simple numeric value (the determinant) that encodes information about the area of our triangle. The elegance of this approach lies in its universal applicability; no matter the shape or orientation of the triangle on the coordinate plane, the formula holds true and yields the correct area.