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Linear combinations involve forming new vectors using scalar multiples of given vectors. This concept helps us understand how vectors can span different geometrical shapes, such as lines, planes, or other higher-dimensional shapes. For any two vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\), any vector of the form \(a\mathbf{v}_1 + b\mathbf{v}_2\) where \(a\) and \(b\) are scalars, is called a linear combination.In our exercise, the span of the vectors \(\mathbf{v}_1 = (2, -2)^T\) and \(\mathbf{v}_2 = (2, -4)^T\) refers to the set of all possible linear combinations of these vectors. Importantly, understanding linear combinations helps us visualize regions in vector spaces, like lines or planes formed by these vectors in \(\mathbf{R}^2\). Linear combinations are the building block for more complex vector operations and understanding geometrical representations in linear algebra.

Linear dependence is a critical concept when determining the span of vectors, as it tells us about the relationships between them. When vectors are linearly dependent, at least one vector can be expressed as a linear combination of others.In our example, \(\mathbf{v}_2 = 2\mathbf{v}_1\). This expression shows that these vectors are linearly dependent. Being dependent means the vectors in question do not span the whole of \(\mathbf{R}^2\) but rather a subset, such as a line.Understanding linear dependence helps identify the minimal number of vectors needed to represent a space. If vectors are dependent, their span is reduced to lower-dimensional geometric objects. Here, instead of spanning a plane, \(\mathbf{v}_1\) and \(\mathbf{v}_2\)'s span is a line, indicating dependence.

Parametric representation is a way to express vectors and the objects they span in a form that depends on one or more parameters, typically scalar variables.For the line spanned by our vectors, we use a parameter, say \(t\), to describe each point on this line. Given that \(\mathbf{v}_1 = (2, -2)^T\), the line's parametric representation is \(t \cdot (2, -2)\) for \(t \in \mathbf{R}\). This describes a continuous set of points along the line.Using parametric representation assists in visualizing how vectors map out lines or planes, especially in problem-solving and geometrical interpretations. It provides a straightforward method to describe all points that make up the vector space's span, giving a clear and concise picture of the geometrical relationships involved.