Linear dependence Definition and 66 Threads

Homework Statement Determine whether the members of the given set of vectors are linearly independent for -\infty < t < \infty. If they are linearly dependent, find the linear relation among them. x(1)(t) = (e-t, 2e-t), x(2)(t) = (e-t, e-t), x(3)(t) = (3e-t, 0) (the vectors are written as...

Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions. τ denotes a linear operator contained in L(V) ι...

Homework Statement show {cos x ,sin x , cos 2x , sin 2x , (cos x − sin x)^2 − 2*sin^2( x)} is not a linearly independent set of real valued functions on the real line R. The Attempt at a Solution Not linearly independent = linearly dependent? So if f(x) = cos (x) g(x) = sin (x) m(x)...

Check for Linear Dependence for: \sin \pi x [-1, 1] I'm thinking it's Linear Dependent. Since it says that any linear combination must be 0. a*x + b*y = 0, a = b = 0. So for any integer x, the value is 0. So [-1, 1] works.

HI. Okay Consider \[\left(\begin{array}{c} 0\\ 0\\ 0\end{array}\right)\] , \[\left(\begin{array}{c} 1\\ 1\\ 0\end{array}\right)\] , \[\left(\begin{array}{c} 1\\ 0\\ 1\end{array}\right)\] , \[\left(\begin{array}{c} 0\\ 1\\ 1\end{array}\right)\] as a subspace of...

(1,0,0) (-3,7,0) and (1,1,0) I'm trying to work out if these vectors are linearly independent or not. Intuitively i believe they are dependent as they span the xy-plane.. but then how do i work out the linear combinations. e.g: (1,0,0) = a(-3,7,0) + b(1,1,0) where a and b are real...

Homework Statement For what values of x are the vectors, [1] [x] [2x] [1] [-1] [-2] [2] [1] [x] linearly dependent? Homework Equations The Attempt at a Solution I made a matrix, [1 ; 1 ; 2 ; 0] [x ;-1 ; 1 ; 0] [2x;-2 ; x ; 0] but I'm having trouble figuring...

Homework Statement The following is from the book Linear Algebra 3rd Edn by Stephen Friedberg, et al: Here aj are scalars of field F and vj are vectors of inner product space V. Homework Equations Theorem 6.3: The Attempt at a Solution Now I don't understand why theorem 6.3...

Hi All A complex equation and its complex conjugate are linearly dependent or independent thanks eman

Homework Statement Given: v_1 = \left(\begin{array}{cc}1\\-5\\-3\end{array}\right) v_2 = \left(\begin{array}{cc}-2\\10\\6\end{array}\right) v_3 = \left(\begin{array}{cc}2\\-9\\h\end{array}\right) For what value of h is v_3 in Span{v_1, v_2} and for what value of h is...

Now I am reading over a theorem, which is very easy to understand, except for a small caveat. Bascally: A set of functions are said to be linearly dependent on an interval I if there exists constants, c1, c2...cn, not all zero, such that c1f1(x) + c2f2(x) ... + cnfn(x) = 0 Well the...

why for bessel equations, if n isn't an integer, you can have the solution y(x)=(c1)Jn(x) +(c2)J(-n)x but isn't true if n's an integer?

Hello everyone, I'm finishing up some matrices review and im' confused on this question i have the matrix: -1 -3 -1 2 5 13 3 -8 3 10 9 -8 1 4 7 -4 I row reduced got this: 1 0 0 3/5 0 1 0 -4/5 0 0 1 -1/5 0 0 0 0 So you can see that this isn't a basis due to column 5 not being 0 0 0...

This may be a really simple proof but its giving me grief. If {v_1, v_2, v_3} is a linearly dependent set of vectors in \mathbb{R}^n, show that {v_1, v_2, v_3, v_4} is also linearly dependent, where v_4 is any other vector in \mathbb{R}^n. Any hints on where to start? I started out by writing...

Suppose that p_0,p_1,p_2...,p_m are polynomials in Pm(F) such that p_j(2)=0 for each j. Prove that (p_0,...,p_m) is not linearly independent in Pm(F). So far I have, suppose that there is a polynomial in the list that is of degree 0, then that polynomial must be 0, hence the list is...

I've already found the answer to this solution but I want to check my methods because the class is very proof-based and the professor likes to take off points for style in proofs on tests: 5. Is {(1, 4, -6), (1, 5, 8), (2, 1, 1), (0, 1, 0)} a linearly independent subset of R^3? Justify your...