Arizona State University (ASU) MAT343 Applied Linear Algebra Exam 2 Practice

Two vectors are considered linearly dependent if one vector can be expressed as a scalar multiple of the other. This principle indicates that there is a straightforward relationship between the two vectors such that they lie along the same line when plotted in a vector space. For instance, if you have two vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \), and there exists a scalar \( k \) such that \( \mathbf{a_2} = k \mathbf{a_1} \), then \( \mathbf{a_1} \) and \( \mathbf{a_2} \) are linearly dependent.

This characteristic contrasts with linearly independent vectors, which cannot be represented as multiples of each other and thus span a space where each vector contributes uniquely to the formation of the vector space. The third option is focused on the intrinsic relationship between the vectors in terms of scalar multiplication, making it the correct understanding of linear dependence.

Other options are not accurate. For example, the assertion that both vectors are non-zero does not imply linear dependence; both could still be independent. Additionally, orthogonality – the condition that two vectors are perpendicular – is a feature of independent vectors. Finally,