Question

The questions are independent. 1. Let F = {(a+2b+c, a+c, a+c, a+2b), a,b,c ∈ R}. Justify that F is a vector subspace of R4, give a base and specify its dimension. 2. Is the family (1, 2, 2), (2, 3, 0), (−1, 3, 1) a basis of R3? Justify.

226

likes
1130 views

Answer to a math question The questions are independent. 1. Let F = {(a+2b+c, a+c, a+c, a+2b), a,b,c ∈ R}. Justify that F is a vector subspace of R4, give a base and specify its dimension. 2. Is the family (1, 2, 2), (2, 3, 0), (−1, 3, 1) a basis of R3? Justify.

Expert avatar
Cristian
4.7
117 Answers
**Question 1:**

Given F = \{ (a+2b+c, a+c, a+c, a+2b) \mid a, b, c \in \mathbb{R} \} is a vector subspace of \mathbb{R}^4 , we need to prove the three conditions for a subspace:

1. F is non-empty: The zero vector (0, 0, 0, 0) is in F when a = b = c = 0 .

2. F is closed under vector addition: Let u = (a_1+2b_1+c_1, a_1+c_1, a_1+c_1, a_1+2b_1) and v = (a_2+2b_2+c_2, a_2+c_2, a_2+c_2, a_2+2b_2) be in F . Then, u + v = (a_1+a_2+2(b_1+b_2)+c_1+c_2, a_1+a_2+c_1+c_2, a_1+a_2+c_1+c_2, a_1+a_2+2(b_1+b_2)) is also in F .

3. F is closed under scalar multiplication: If u = (a+2b+c, a+c, a+c, a+2b) is in F and k is a scalar, then ku = (ka+2kb+kc, ka+kc, ka+kc, ka+2kb) is in F .

Therefore, F is a subspace of \mathbb{R}^4 .

To find a basis for F , we take a = 1, b = 0, c = 0 , a = 0, b = 1, c = 0 , and a = 0, b = 0, c = 1 , resulting in the basis:

\{(1, 1, 1, 1), (0, 2, 0, 2), (1, 1, 1, 0)\}

**Answer:**

The vectors (1, 1, 1, 1), (0, 2, 0, 2), (1, 1, 1, 0) form a basis for F , and the dimension of F is 3.

---

**Question 2:**

To determine if (1, 2, 2), (2, 3, 0), (-1, 3, 1) is a basis for \mathbb{R}^3 , we check for linear independence by row-reducing the matrix formed by these vectors.

The row-reduced echelon form of the matrix composed of these vectors is the identity matrix, indicating that the vectors are linearly independent and span \mathbb{R}^3 .

**Answer:**

The vectors (1, 2, 2), (2, 3, 0), (-1, 3, 1) form a basis for \mathbb{R}^3 .

Frequently asked questions (FAQs)
What is the hyperbolic cosine of 2π?
+
What is (2^3)^4? Simplify the expression and provide the answer in exponential form.
+
What is the derivative of cos(2x) - sin(3x) + tan(4x) ?
+
New questions in Mathematics
A car tire can rotate at a frequency of 3000 revolutions per minute. Given that a typical tire radius is 0.5 m, what is the centripetal acceleration of the tire?
two particles start at the origin and move along the x axis. for 0 <= t <= 10, their respective position functions are given by x1 = cos(t) and x2 = (e^-3t) + 1. for how many values of t do the particles have the same velocity?
If L (-2, -5) reflected across y = -4. What are the coordinates of L?
If L = (-2, -5) is reflected across y= -4 , what are the coordinates of L?
How do you think the company has increased or decreased its income?
Determine the equations of the recipes that pass through the following pairs of points P1 (2;-1) and p2 (4;-1)
Additionally, the boss asked Armando to determine how many toy sales branches he would have in the fifteenth year, knowing that the first year they started with two branches, by the second they already had 5 branches and, by the third year, they had 8 branches. From the above, determine the number of branches it will have for the fifteenth year.
A, B, C and D are numbers; If ABCD = 23, What is the result of ABCD BCDA CDAB DABC operation?
What’s 20% of 125?
logy/logx + logz/logy + logt/logz = 8x².t x=?
Suppose the Golf ball market is perfectly competitive and the functions are known: Q = 120 – 2Px – 2Py 0.2I Q = 2Px 40 Where I = Consumers' income ($200) and Py = Price of Good Y (40) Calculate the equilibrium elasticity: a) 1.6 b) -6 c) 6 d) 0.6
A study reports the following final notation: F (3, 32) = 9.50, p < .05. How many total participants were involved in this study? Group of answer choices 34 32 36
What’s the slope of a tangent line at x=1 for f(x)=x2. We can find the slopes of a sequence of secant lines that get closer and closer to the tangent line. What we are working towards is the process of finding a “limit” which is a foundational topic of calculus.
User The average height of Aranka, Böske, Cili, Delinke and Lili is 172 cm. We know that Aranka and Cili are both 172 cm tall. The sum of the heights of Böské and Delinke is 336 cm. How tall is Lili?
392929-9
2x-5-x+2=5x-11
P 13. Let P a point inside of a square ABCD. Show that the perpendicular lines drawn from A, B, C, respectively D, to BP, CP, DP, respectively AP are concurrent. Use geometric rotation.
How much does 7.2 moles of ammonium dichromate weigh? (NH4)2Cr2O7
Sally’s sales for last Sunday were $1,278. That was an increase of 6.5% over her sales for the previous Saturday. What were her sales for the previous Saturday?
Solve the system of equations by the addition method. 0.01x-0.08y=-0.1 0.2x+0.6y=0.2