Answered: Let P = {(x, y, z) E R³|x – y + 3z = 0}…

Let P = {(x, y, z) E R³|x – y + 3z = 0} Q = {(x, y, z) E R³| x + 3y – z = 1} (i) Show that P is a subspace of R°. (ii) Show that Q is not a subspace of R°.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.3: Subspaces Of Vector Spaces

Problem 56E: Give an example showing that the union of two subspaces of a vector space V is not necessarily a...

See similar textbooks

Related questions

Q: Determine if (x, y, z, t) ∈ R^4 such that y = −x and z = 0, and t = 2x form a subspace of R^4

A: Use the necessary and sufficient condition for a subspace of a vector space

Q: Let X = 1 ,and let V = {A E M2(R) | AX = XA}, a subspace of M2(R). 0. Find a basis for V, and write…

Q: Let W = {a + bx + cx² + dx³[ c – 3d = 0 } be a subspace of P3. Then the dimension of W is equal to O…

Q: Let H be the set of points in the second quadrant in the plane V=R^2. That is, H = { (x,y) | x ≤ 0,…

Q: (a) find the orthogonal complement S⊥, and (b) find the direct sum S⊕S⊥. S is the subspace of R3…

Q: Let A = { | 2.x |x € R}. Is A a subspace of R³? З Let B x + y + z = 0}. Is B a subspace of R³?

Q: f(x) = 9, g(x) = 7x – 4 and h(x) = -2x². Consider the inner product (p, q) = / p(z)q(x) dæ in the…

A: Let S={v1,v2,v3} represent the orthonormal basis for the subspace. By Gram Schmidt Process,…

Q: if a vector space is the set of real valued continuous functions over R , then show dy d y -9. dx?…

Q: If V is the vector space of real valued continuous functions , then show that the set W of all d’y…

A: Our Aim is to show that the set W of all solutions of the differential equation3d2ydx2-5dydx+7y=0…

Q: Find the basis for the given subspace of R°, and sta its dimensions for the plane 2x – 4y +5z =0. 1…

Q: Let V be the set of vectors (x, y, z) E R° such that r(y² + z2) = 0. Is V a subspace of R³?

A: 16. Let V be the set of vectors x, y, z∈ℝ3 such that xy2+z2=0 We have to check whether V is a…

Q: Find a basis for the given subspace of R’, and state its dimension for the plane 4 x - 2 y + 5 z = 0…

A: Solution:-

Q: :) Let T : U = р — 2q — 0, р, 9, r, s E R Is U a subspace of M2.2(R), if so, find its basis and…

Q: Let R be the field of real numbers. Then {(x,2y,3z): x, y, z ER}is a subspace of R°(R) ?

Q: 1 -1 Find the distance from y to the subspace spanned by u. U= y= NO N1

A: Here given vectors are: u=2021y=1-11-1

Q: Let S be the subspace of R2 spanned by e1 and let T be the subspace of R2 spanned by e2. Is S ∪ T a…

Q: Let B = {1, x, eX, xe*} be a basis for a subspace W of the space of continuous functions, and let Dx…

Q: Let V = P2(t) with inner product defined by (f, g) = So f(t)g(t) dt. Find a basis for the subspace W…

A: An inner product is a generalization of the dot product. A real vector space, an inner product…

Q: Find a basis for the given subspace of R , and state its dimension for the plane 2 x - 4 y+ 5 z = 0.…

Q: For a certain subspace W, the orthogonal projection of the vector y = 3 onto W is ŷ 2. Use this…

Q: 16. Let V be the set of vectors (x, y, z) E R³ such that r(y? + z²) = 0. Is V a subspace of R?

Q: 16. Let V be the set of vectors (x, y, z) E R³ such that r(y?+ z²) = 0. Is V a subspace of R³?

A: Test for subspace for V to be a subspace If u and v are elements of v then u +av must be in V where…

Q: About the sets, S1 = {(x,y) ∈ R2: x = y3} S2 = {(x,y,z) ∈ R3: x = (y−z)2} S3 = {(x,y) ∈ R2: x4 +…

Q: Let W = (a + bx+ cx + dx'|a+ b= 0 and c -3d = 0 ) be a subspace of P Then the dimension of W is…

Q: Determine whether U = {(1, s, t)ls,t e R} is a subspace of R.

A: FOR ANY SUBSET U TO BE A SUBSPACE OF ANY VECTOR SPACE V IT IS NECESSARY THAT U HAS ZERO VECTOR IN…

Q: Is T = {(x,y,z)| x+2 y+3z=0} a subspace of R?

A: Subspace

Q: Let W = {(0, x, y, z): x - 6y + 9z = 0} be a subspace of R. Then a basis for W is: %3D

Q: Let W = (a + bx + cx? + dx'la + 2c 0 and b -d 0} be a subspace of P, %3D Then dimension of W is…

Q: Suppose that U cr|xyer F4 y EF, U is a subspace of F4- you don't have to! Find a subspace V of F4…

Q: Suppose U = {(x, y,x + y,x – y, 2x) e F° : x, y e F}. Find a subspace W of F such that F5 = U + W.

Q: Is W = ((x,y) | xy=0} a subspace of R??

Q: Let W = {(x, y, 1) | x, y, E R}. Is W a subspace of R3?

Q: For a fixed a ϵ R, determine the dimension of the subspace of Pn(R) defined by {f ϵ Pn(R): f(a) =…

A: Assume a ϵ R and the set V=f∈PnR: fa=0 is subspace. Now, assume the arbitrary polynomial f from V.…

Q: If v is in R°, then the set of vectors a with v x a = 0 is a subspace.

Q: Consider the set S = yl E R : 4x +6z=7y, x – 13y=0 . Is this a subspace of R ? Explain why or why…

Q: Let W be the set of all vectors of the form b. 3c - 2 b C where b and ceR. Find vectors u and v such…

A: Spanning set

Q: Is Y = (x,y,z) | y=z} a subspace of R?

Q: ) The degree of pa is equal to the dimension of the cyclic subspace Z (a ; T) .

Q: Find the basis for the given subspace of R°, and state its dimensions for the plane 2x – 4y +5z =0.…

Q: 3) Show that the set of all elements of the form (x + 2y,x + 3z, 2x + y + z), where x, y, z E F, is…

A: We have, (x+2y , x+3z , 2x+y+z),where x, y, z € F.

Q: Let S be the set of all elements of the form (x + 2y , y , -x + 3y) in R3 , where x,y belongs to R…

Q: W e R"|2, > 0 over the field of reals is a subspace of the specified vector space ER"

A: Given that W=x1x2x3⋮xm∈Rm|x1≥0 and V=x1x2x3⋮xm∈Rm The objective is to show that W is subspace of V.…

Q: 2. The subspace (a) 2y x+y+z=0 1 in R3 is a plane whose equation is x-2=0 (b) y (c). x-2y+z=0 Py27.…

Q: Prove that S = {(x, y, z) E R³: 2x + 3y+z = 0} is a subspace of the vector space R3.

A: Explanation of the answer is as follows

Q: Let V - P2 be the vector space of polynomials of degree at most 2, and let H be the span of the…

Q: Let V=R^3 and let H be the subset of V of all points on the plane -3x+5y-5z=0. Is H a subspace of…

Q: 2. Let U = {p E P3(R)| p(3) = 0} (polynomial of degree at most 3 with real coefficients). Show that…

A: Given U=p∈P3R|p3=0 We have to show that U is a subspace of P3R. (1) Let P1,P2∈U then P13=0 and P23=0…

Q: Let W be the subspace spanned by u, and u,. Write y as the sum of a vector in W and a vector…

A: There are two vectors u1 and u2 is given. write the vector y as the sum of a vector in W and a…

Q: Let W = {a + bx + cx2 + dx³| a + b = 0 and c – 3d = 0 } be a subspace of P3. Then the dimension of W…

Q: a) Let X = {fe C[0,1]:f (0) = 0} Y = {g ex :[g (t) dt = 0} %3D Prove that Y is a proper subspace of…

A: Note: According to Bartleby guidelines; for more than one question asked; only the first one is to…

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Let
P = {(x, y, z) E R³|x – y + 3z = 0}
Q = {(x, y, z) E R³| x + 3y – z = 1}
(i) Show that P is a subspace of R°.
(ii) Show that Q is not a subspace of R°.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 34. ,
Proof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.
Calculus Repeat Exercise 45 for B={e2x,xe2x,x2e2x}. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.
Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 37. V = P, W is the set of all polynomials of degree 3