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The reason why is that the first vector describes the line y 2x y 2 x and the second vector describes the line 2y 4x 2 y 4 x.
Linearly independent linear algebra math. Therefore any set consisting of a single nonzero vector is linearly independent. The third vector is unneeded as a basis for r2. These two vectors are not linearly independent. Any set of two of those vectors by the way are linearly independent.
Note that this geometric intuition will only work if you are only checking if two functions are linearly independent like the intuition you wrote for mathbb r 2 for example three vectors in mathbb r 2 may not be mutually parallel but they must form a linearly dependent basis. Putting a third vector in to a set that already spanned r2 causes that set to be linearly dependent. If we were to add any scalar multiple of these two vectors you would still get another vector that describes the line y 2x y 2 x. The rank of a matrix or a linear transformation is the dimension of the image of the matrix or the linear transformation corresponding to the number of linearly independent rows or columns of the matrix or to the number of nonzero singular values of the map the rank of a matrix is implemented as matrixrank m.
On the contrary if at least one of them can be written as a linear combination of the others then they are said to be linearly dependent. Linear algebra is actually a deeply interesting branch of math but when we mindlessly gaze for hours at matrices we don t get an understanding we get eye strain. Endgroup projectilemotion 2 days ago. In fact including 0 in any set of vectors will produce the linear dependency 0 0v 1 0v.
A set consisting of a single vector v is linearly dependent if and only if v 0.
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