Question

Asked Dec 27, 2019

14 views

The vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1) form a basis for F4.

Find the unique representation of an arbitrary vector (a1, a2, a3, a4) in F4 as a linear combination of u1, u2, u3, and u4.

Step 1

Given,

u_{1} = (1, 1, 1, 1), u_{2} = (0, 1, 1, 1), u_{3} = (0, 0, 1, 1), and u_{4} = (0, 0, 0, 1)

Consider the linear combination,

(a_{1}, a_{2}, a_{3}, a_{4}) = xu_{1}+ yu_{2}+zu_{3}+tu_{4} …(1)

Then,

(a_{1}, a_{2}, a_{3}, a_{4}) = x (1, 1, 1, 1) + y (0, 1, 1, 1) + z (0, 0, 1, 1) + t (0, 0, 0, 1)

Now add the corresponding elements as follows,

(a_{1}, a_{2}, a_{3}, a_{4}) = [x , (x+y) , (x+y+z) , (x+y+z+t)]

Step 2

Now equate the coordinates of both sides

a1 = x

b2 = x+y …(2)

c3 = x+y+z …(3)

d4 = x+y+z+t …(4)

Substitute the value of x in (2)

b2 = a1+y implies y = b2-a1

Substitute the value of x and y in (3)

c3 = a1+(b2-a1)+z

...

Tagged in

Find answers to questions asked by student like you

Show more Q&A

Q: a. Show that the points (2, 9), (−1, −6), and (−4, −3) are not collinear by finding the slope betwee...

A: a) Use the slope formula to find the slope between (2, 9) and (−1, −6)

Q: Match the following functions with the graph. (a) The cost of painting a wall as a function of its s...

A: Part (a):The cost of painting a wall as a function:According to the statement,The cost will be incre...

Q: Two or more linear equations taken together make up a __________ of linear equations.

A: Two or more linear equation taken together make up a system of linear equation.Answer – system

Q: A theater charges $50 per ticket for seats in Section A, $30 per ticket for seats in Section B, and ...

A: A theater charges $50 per ticket for seats in Section A, $30 per ticket for seats in Section B, and ...

Q: For Exercise, a. Write the equation of the hyperbola in standard form. b. Identify the center, verti...

A: (a) Write the equation of the hyperbola in the standard form as follows.

Q: Explain why the f(x) = x^2 is not an exponential function.

A: To explain why the f(x) = x^2 is not an exponential function.

Q: For Exercise, determine the standard form of the parabola subject to the given conditions. Vertex: (...

A: To determine the standard form of the parabola.

Q: For Exercise, determine the standard form of an equation of the parabola subject to the given condit...

A: Plot the given points line joining focus and vertex is parallel to y axis. That means axis is parall...

Q: What is meant by a proper rational expression?

A: A rational expression is proper if the degree of the numerator is less than the degree of the denomi...