Answered: Suppose that U and V are both subspaces…

Suppose that U and V are both subspaces of R . Prove that their intersection U ∩ V = {v : v ∈ U and v ∈ V } is also a subspace of R .

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.5: Subspaces, Basis, Dimension, And Rank

Problem 10EQ

See similar textbooks

Related questions

Q: Let V be an inner product space and u a ﬁxed vector in V . Prove that the set of all vectors in V…

A: The objective is to prove that the set of all vectors in V that are orthogonal to u is a subspace…

Q: U is a subspace of Rn, and S is a linearly independent subset of U, then span(S) = U.

A: False

Q: Show that W = {(x1, x2): x1 ≥ 0 and x2 ≥ 0}, with the standard operations, is not a subspace of R2.

Q: Let V = M2x2. W. (R) = € V : a, b, cER W2(R) € V : a € R then prove that W, and W2 are subspaces of…

A: Necessary condition to be a subspace is: Vector addition property Scalar multiplication property.…

Q: Let W, and W2 be subspaces of a vector space V. Prove that V is the direct sum of W, and W2 if and…

A: First assume that V is the direct sum of W1 and W2 W1+W2=V and W1∩W2=0 We have to prove that each…

Q: Let (X, F) be a space. Prove that a subspace of a subspace is a subspace of (X, F).

A: Here given X, I is a topological space.

Q: Let U and W be subspaces of V over a ficld F. Show that the union U UW is a subspace if and only if…

Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

Q: Prove that if M is a closed subspace and N is a finite dimensional subspace of a normed space X,…

A: This is a problem of Functional Analysis.

Q: Show if W is a subspace of the vector space X, then W is convex.

A: Definition used: A set K in a vector space V is called convex if and only if x, y belongs to K,…

Q: If A is a normed subspace of E, proof that bd(A) is a subspace of X.

A: Please check the answer in the next step

Q: Let A be a fixed m × n matrix. Prove that the set W = {x ∈ Rn: Ax = 0} is a subspace of Rn.

Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Let V be a finite-dimensional vector space, and let U, W be subspaces of V. Prove that (U + W)° C…

Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

Q: Prove that the set ? = {(a + b, a -b): a and b are real numbers} is a subspace of ℝ2 with the…

Q: Prove that the subspaces of R3 are {0}, R3 itself, and any line or plane passing through the origin.

A: Let S be a subspace for R3 Hence, If S is the subspace then by definition:…

Q: 4. Use the definition of a subspace to prove that the set V:= :, y, z E R, y+ : 0 is a subspace of…

Q: Determine whether the set A= (x,y,z):x+y%3D3Z is a subspace of R or not?

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

Q: Let W be a subspace spanned by the u's, an

A: Given, W be a subspace spanned by u's, and write y as the sum of a vector in W and a vector…

Q: Prove that if U and V are subspaces of R" and U <V then V+<U+.

Q: Suppose U and W are two-dimensional subspaces of R3. Show that U∩W≠{0}

A: We want to show that U∩W≠{0}

Q: Determine whether W is a subspace of V V = Mnn, W is the set of diagonal n X n matrices

Q: Let W be a subset of the vectors in R where W = X2 X4 -x3 = x, - x, }. Show that W is a X3 3. X4…

A: We will use the formal definition of a subspace of a vector space to prove that, W is a subspace of…

Q: Determine whether W is a subspace of V = C, where W = {(0, b,c) E C³

A: V=ℂ3 and W=0,b,c∈ℂ3

Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that , spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: Let W be a subspace of Rn. w ∩ w⊥ = {o}

Q: Let S be the collection of vectors [] in R² that satisfy the given property. In each case, either…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

Q: Show that W = {(a, 0, b)la, b = R} is a subspace of R³.

A: As per guidelines we are supposed to solve the maximum first questions only .kindly post another…

Q: If V is a vector space and W is a subset of V that contains the zero vector then W must be a…

A: Given statement If V is a vector space and W is a subset of V that contains the Zero vector and W…

Q: Show that the set W={(x,y,1)|x and y are real} is not subspace of R3.

Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

Q: Take z in W", and let u represent any element = is in wt) Take z, and z in w", and let u be any…

A: Introduction: There are two vectors x,y∈Rn, if they are orthogonal or perpendicular x·y=0. Notation:…

Q: If x is in W, then x is orthogonal to some vector in a subspace W

A: SOLUTION-

Q: a+b+2c 2a +2b+4c+d :a,b,c, d e R b+c+d За + Зс + d

Q: If W is a subspace of R" and ax is in W, where a is a nonzero scalar and x is in R". Then x is in W/…

A: We have to check

Q: Let A be an n x n matrix. Verify that V = {x e R" : Ax = 3x} is a subspace of R".

Q: If V is any three-dimensional subspace of R°, then V has infinitely many bases.

Q: - Let V = R and W = {(a, b, c) E V \a + b = c}. Is W a subspace of V? If so, what is its dimension?

Q: Determine whether the given set SS is a subspace of the vector space V. (2,3,4,5)

A: For subspace test , if u and v is any element of S then au+v must be in S where a is any scalar

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: Determine: if W = the set of all vectors in R2 whose components are integers is a subspace.

Q: Let V and W be vector spaces with subspaces V1 and W1, respectively. If T : V --> W is linear,…

A: Given: T:V →W is a linear transformation, V1, W1 are respectively subspaces of V and W

Q: Let S be a subspace of R" and let S be its orthogonal complement. Prove that S is also a subspace of…

Q: i) Let V = M2x2(R) and %3D W; = {(: :) ev:a.hceR} and W2 = {(". :) ev :a,be R Prove that W1 and W2…

Q: If V is a 13 dimensional subspace of R20 then the dimension of V- must be

A: Definition of Orthogonal Complements Let V be a subspace of ℝn. Its orthogonal complement is the…

Q: x=y=z

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

Suppose that U and V are both subspaces of R . Prove that their intersection

U ∩ V = {v : v ∈ U and v ∈ V }

is also 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

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

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

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 34. ,
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=Mnn,WAinMnn:detA=1
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[a0a]}
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
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba+b+1]}
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba]}
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=Mnn, W is the set of diagonal nn matrices
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=M22,W={[abcd]:adbc}
Consider the vector spaces P0,P1,P2,...,Pn where Pk is the set of all polynomials of degree less than or equal to k, with standard operations. Show that if jk, then Pj is the subspace of Pk.
Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
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.
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 39.