Question

Consider the two vector valued functions x1(t) = (2e^t , 3) and x1(t) = (4, 6e^-t...

Consider the two vector valued functions x1(t) = (2e^t , 3) and x1(t) = (4, 6e^-t ). For any given fixed value t0, show that the two dimensional vectors x1(t0) and x2(t0) are linearly dependent. At the same time, show that x1 and x2 as functions of t are linearly independent.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Consider the two vector-functions listed below. Justify your answers for each question. x1(t) = [t, 2t]...
Consider the two vector-functions listed below. Justify your answers for each question. x1(t) = [t, 2t] and x2(t) = [t , t^2] (a) Are the vector-functions x1(t) and x2(t) linearly independent on the interval (−∞,∞)? (b) Does there exist a point t0 such that the constant vectors x1(t0) and x2(t0) are linearly dependent? (c) Can the vector-functions x1(t) and x2(t) be solutions to a first-order homogeneous linear system? DIFF. EQUATIONS
Find the vectors T and N and the binormal vector B = T ⨯ N, for...
Find the vectors T and N and the binormal vector B = T ⨯ N, for the vector-valued function r(t) at the given value of t. r(t) = 6 cos(2t)i + 6 sin(2t)j + tk,    t0 = pi/4 find: T(pi/4)= N(pi/4)= B(pi/4)=
Let X = (X1, X2) T be a two-dim random vector, and (R, Θ) be its...
Let X = (X1, X2) T be a two-dim random vector, and (R, Θ) be its polar coordinates, i.e. X1 = R cos(Θ) and X2 = R sin(Θ). Show that X is spherically symmetric if and only if R and Θ are independent, and Θ ∼Uniform(0, 2π).
(a) Consider x^2 + 7x + 15 = f(x) and e^x = g(x) which are vectors...
(a) Consider x^2 + 7x + 15 = f(x) and e^x = g(x) which are vectors of F(R, R) with the usual addition and scalar multiplication. Are these functions linearly independent? (b) Let S be a finite set of linearly independent vectors {u1, u2, · · · , un} over the field Z2. How many vectors are in Span(S)? (c) Is it possible to find three linearly dependent vectors in R^3 such that any two of the three are not...
Consider the vector a(t)=〈cost,sint〉 with components that depend on a real number t. As the number...
Consider the vector a(t)=〈cost,sint〉 with components that depend on a real number t. As the number t varies, the components of a(t) change as well, depending on the functions that define them. Write the vectors a(0) and a(π) in component form. Show that the magnitude ∥a(t)∥ of vector a(t) remains constant for any real number t. As t varies, show that the terminal point of vector a(t) describes a circle centered at the origin of radius 1.  
Find the standard matrix for the following transformation T : R 4 → R 3 :...
Find the standard matrix for the following transformation T : R 4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 + 4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and T(~e4). (b) Find an equation in vector form for the set of vectors ~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of T?
Problem 2. (20 pts.) show that T is a linear transformation by finding a matrix that...
Problem 2. (20 pts.) show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2, ... are not vectors but are entries in vectors. (a) T(x1, x2, x3, x4) = (0, x1 + x2, x2 + x3, x3 + x4) (b) T(x1, x2, x3, x4) = 2x1 + 3x3 − 4x4 (T : R 4 → R) Problem 3. (20 pts.) Which of the following statements are true about the transformation matrix...
Let A be a given (3 × 3) matrix, and consider the equation Ax = c,...
Let A be a given (3 × 3) matrix, and consider the equation Ax = c, with c = [1 0 − 1 ]T . Suppose that the two vectors x1 =[ 1 2 3]T and x2 =[ 3 2 1] T are solutions to the above equation. (a) Find a vector v in N (A). (b) Using the result in part (a), find another solution to the equation Ax = c. (c) With the given information, what are the...
Enlarge the following set to linearly independent vectors to orthonormal bases of R^3 and R^4 {(1,1,1)^t,...
Enlarge the following set to linearly independent vectors to orthonormal bases of R^3 and R^4 {(1,1,1)^t, (1,1,2)^t} could you show me the process, please
Determine whether the given set ?S is a subspace of the vector space ?V. A. ?=?2V=P2,...
Determine whether the given set ?S is a subspace of the vector space ?V. A. ?=?2V=P2, and ?S is the subset of ?2P2 consisting of all polynomials of the form ?(?)=?2+?.p(x)=x2+c. B. ?=?5(?)V=C5(I), and ?S is the subset of ?V consisting of those functions satisfying the differential equation ?(5)=0.y(5)=0. C. ?V is the vector space of all real-valued functions defined on the interval [?,?][a,b], and ?S is the subset of ?V consisting of those functions satisfying ?(?)=?(?).f(a)=f(b). D. ?=?3(?)V=C3(I), and...