how to determine whether a set of vectors is linearly independent?
63734
We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero.
754456
The following results from Section 1.7 are still true for more general vectors spaces. A set containing the zero vector is linearly dependent. A set of two vectors is linearly dependent if and only if one is a multiple of the other. A set containing the zero vector is linearly independent.
479088
A scalar signal contains a single element. The signal could be a one-dimensional array with one element, or a matrix of size 1-by-1. A vector signal contains one or more elements, arranged in a series.
834753
If, on the other hand, there exists a nontrivial linear combination that gives the zero vector, then the vectors are dependent. Example 2: Use this second definition to show that the vectors from Example 1— v 1 = (2, 5, 3), v 2 = (1, 1, 1), and v 3 = (4, −2, 0)—are linearly independent.
952431
Conversely, if your matrix is non-singular, it's rows (and columns) are linearly independent. Matrices only have inverses when they are square. ... This means that if you want both your rows and your columns to be linearly independent, there must be an equal number of rows and columns (i.e. a square matrix).
549242
Linear Combination of Vectors. If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors. For example, suppose a = 2b + 3c, as shown below. ... Thus, a is a linear combination of b and c.
257458
If uTv=0 then u and v are orthogonal. The null space of A is the set of all solutions x to the matrix-vector equation Ax=0. ... then the span of v1 and v2 is the set of all vectors of the form sv1+tv2 for some scalars s and t.
92762
['It should be easy to understand.', 'It should be simple to compute.', 'It should be based on all the items.', 'It should not be affected by extreme values.', 'It should be rigidly defined.', 'It should be capable of further algebraic treatment. Share this. Share on FacebookTweet on TwitterPlus on Google+']
158974
['example. R = rref( A ) returns the reduced row echelon form of A using Gauss-Jordan elimination with partial pivoting.', 'R = rref( A , tol ) specifies a pivot tolerance that the algorithm uses to determine negligible columns.', '[ R , p ] = rref( A ) also returns the nonzero pivots p .']
258019
Independence in systems of linear equations means that the two equations only meet at one point, and the solution is the intersection of the two lines. ... If the slopes are the same, then the system is either dependent (same line with many solutions) or the lines are parallel (inconsistent system with no solution).
1391243
An SVM is a non-parametric classifier that finds a linear vector (if a linear kernel is used) to separate classes. Actually, in terms of the model performance, SVMs are sometimes equivalent to a shallow neural network architecture.
923392
Theorem 6.1: A matrix A is invertible if and only if its columns are linearly independent. ... If A is invertible, then its columns are linearly independent.
174180
In three-dimensional space, vector →A has three vector components: the x-component →Ax=Ax^i A → x = A x i ^ , which is the part of vector →A along the x-axis; the y-component →Ay=Ay^j A → y = A y j ^ , which is the part of →A along the y-axis; and the z-component →Az=Az^k A → z = A z k ^ , which is the part of the ...
627596
(1) T is one-to-one if and only if the columns of A are linearly independent, which happens precisely when A has a pivot position in every column. (2) T is onto if and only if the span of the columns of A is Rm, which happens precisely when A has a pivot position in every row. Example.
964695
One more definition: Two functions y 1 and y 2 are said to be linearly independent if neither function is a constant multiple of the other. For example, the functions y 1 = x 3 and y 2 = 5 x 3 are not linearly independent (they're linearly dependent), since y 2 is clearly a constant multiple of y 1.
165785
So, in your case, W(sinx,cosx)=sinx⋅(−sinx)−cosx⋅cosx=−sin2x−cos2x=−1≠0 for any I, so sinx and cosx are linearly independent on any interval I.
439289
Answer: Given two unit vectors, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel.