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Vector Algebra
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
It is geometrically evident that (ABC) is positive or negative according to whether the vectors A, B, C form a right-handed or a left-handed triple. When (ABC) ≠ 0, the vectors are said to be linearly independent; otherwise, the vectors are said to be linearly dependent. It follows that three nonzero vectors are linearly dependent if and only if they are coplanar. In the case of two vectors, linear dependence implies that the vectors are scalar multiple of each other. Hence two nonzero vectors are linearly independent if and only if the vectors are not parallel
Linear Dependence and Independence
Published in Ravi P. Agarwal, Cristina Flaut, An Introduction to Linear Algebra, 2017
Ravi P. Agarwal, Cristina Flaut
The concept of linear dependence and independence plays an essential role in linear algebra and as a whole in mathematics. These concepts distinguish between two vectors being essentially the same or different. Further, these terms are prerequisites to the geometrical notion of dimension for vector spaces.
Matrices and linear transformations
Published in Alan Jeffrey, Mathematics, 2004
The task of determining the rank of a matrix A may be greatly simplified by using the fact that the addition of a multiple of one row of A to another will not alter the linear dependence of the rows of A, and so will not alter rank A. This property may be used to simplify the structure of A until its rank becomes obvious. The simplification is accomplished by reducing A by successive row operations of this type to what is called a row-equivalent echelon matrix, in which all elements below a line drawn through the leading diagonal of the original elements a11, a22, a33, … are zero.
Linear independence from a perspective of connections
Published in International Journal of Mathematical Education in Science and Technology, 2022
Hamide Dogan, Edith Shear, Angel F. Garcia Contreras, Lion Hoffman
The remaining five participants brought out slightly different thought processes for their use of the MFC. That is, these participants worked with columns of matrices (specifically focusing on rref form) or component values of vectors. Below, SC6 gives us one such response. In the response, he is considering a linear independence idea through the impossibility of writing component values of one vector in terms of component values of other vectors of a set. He does this with a geometric view of the scalar multiplication operation making vectors ‘longer or shorter’ and/or changing ‘orientation’. SC6:the definition of linear independence is a system a set of vectors that cannot be expressed by a linear combination [of vectors of the set] … so, my linear independence would basically tell me that I have to multiply this [vector {1,2}] by a scalar to get this one [{3,4}]. But the problem is the components between the two are not the same. Multiplying it by a scalar would only I can only make it longer or shorter, but I cant change the orientation. So, multiplying this one [{1,2}] by just a scalar multiple I can never get the second vector [{3,4}]. Participant SB15, too, applied a geometric view, describing a linear combination process as ‘this vector is a combination of the other vectors because it would go a little bit in the x [x-axis], a little bit in the y [y-axis] and a little bit in the z [z-axis]’.
Laboratory investigation of the breaking wave characteristics over a barrier reef under the effect of current
Published in Coastal Engineering Journal, 2019
Yu Yao, Wenrun He, Zhengzhi Deng, Qiming Zhang
Figure 8(a) also lists the predictions from the fitted Equation (4) for all tested wave conditions with each current flowrate . The for each dataset with the same is generally larger than 0.75 (not shown), suggesting that Equation (4) can be satisfactorily applied to describe despite of the presence of current. The fitted values of the parameters in Equation (4), i.e. and , are plotted against , respectively, in Figure 8(b). A linear dependence on can be found for both and , which gives