Determine which of the following sets are subspaces. {[: ] a b E R²x2 | ad – bc d W1 W2 = {(x, y, z, t) E R* | 2x + 3y = z – t} Justify your answer !!! Please be reminded that IR2×2 is the vector space of 2 x 2 matrices, and, Rª is the vector space of all 4-tuples...

Determine which of the following sets are subspaces. {[: ] a b E R²x2 | ad – bc d W1 W2 = {(x, y, z, t) E R* | 2x + 3y = z – t} Justify your answer !!! Please be reminded that IR2×2 is the vector space of 2 x 2 matrices, and, Rª is the vector space of all 4-tuples...

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 5CM: Take this test to review the material in Chapters 4 and 5. After you are finished, check your work...

See similar textbooks

Related questions

Q: 2. Let V be a vector space over R, and let S1 and S2 be subspaces of V. (a) Prove that Si C S2 =…

A: I have solved by assuming basis of the givem subspaces

Q: Let S = { (x, y, z, w) E R*; x – 2y + w = 0 and 2x + z – w = 0 } . - Prove that S is a subspace by…

Q: Let V = M2×2(F), be the vector space of all 2 × 2 matrices over the real number field. Define W1 =…

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 P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

A: We first write given vectors is matrix form and then reduce the given matrix in row echelon form.…

Q: 2. Determine whether or not the following subsets of the given vector spaces, are subspaces: (a) The…

Q: Which of the following subset is the subspace of the specified vector space? I. T = {(x,0, z,0) | x,…

Q: (b) Let V = M2x2(F), be the vector space of all 2 x 2 matrices over the real number field. Define W;…

A: V=M2×2(F), a vector space of 2×2 matrices And F=ℝ, fieldGiven: W1=abca∈V: a, b, c∈F…

Q: 3. Determine whether the following are subspaces of the given vector space V. a. S= {ax? + bx + c a+…

Q: Which of the followings is not a subspace of the vector space V? a) V = Mxn(R),W = {A € VIAT = A} b)…

Q: Given U_1, ..., U_m, a collection of subspaces of some vector space V, recall that U_1 + ... + U_m…

A: Here, in the question, there is a collection of subspaces given. We have to give an example of the…

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

Q: Determine whether the given set S is a subspace of the vector space V. ? V 1. V = RX", and S is the…

Q: Determine whether the given set S is a subspace of the vector space V. DA. V = R², and S is the set…

A: As per our company guidelines, we are supposed to solve first question for you. Please submit other…

Q: Let H be a subspace of a vector space V, and suppose that V has dimension d. Which of the following…

A: a) Given a vector space V, let S=v1, ..., v2 be a basis for V. Consequently dimV=d. Now given H, a…

Q: Verify that W is a subspace of V. In each case, assume that V has the standard operations.{(x, y, 4x…

Q: Either prove or disprove that the set V is a subspace V ⊂ R^n defined by V={x: Ax=λx}where A is…

Q: Which of the following subsets S of a vector space V are subspaces? O A. V = R?, S = {(x1,0) : x1 e…

Q: The set R' of all column vectors of length three, with real entries, is a vector space. Determine…

Q: . In the following either prove that the set W is a subspace of the vector space V or give a…

Q: and M(R}3«3, 2 ×2 being the set of all matrices with real variables U- {x e M(R}** : AX = x A}…

Q: Let W1 and W2 be subspaces of a vector space V over K, and let S1 be a subset of Wı and S2 a subset…

A: # we are entitled to solve one question at a time , please resubmit the other question if you wish…

Q: Which of the following subset is the subspace of the specified vector space? I. T = {(x,0, z,0) | x,…

Q: Determine which (if any) of the following sets U, V,W are subspaces of R". For those that are,…

Q: 10. Let V = M2x2; W, (R) = {|; ev {: ]ev EV: а, b, сER {I°% ev : a €R W2(R) then prove that W1 and…

A: A set W is said to be subspace of vector space V over F if W itself form a vector space over field F…

Q: 2. Prove the following given subsets of the vector space, M23, of all 2 x 3 matrices is a subspace.…

Q: Determine whether the given set S is a subspace of the vector space V. OF. V = R"xn, and S is the…

Q: Be F1(x, x, x) e R³ and F2(a, b,0) e R³ a)show that F1 and F2 are subspaces of R b)show that R° = F1…

Q: Determine whether the given set S is a subspace of the vector space V. v 1. V = Rnxn¸and S is the…

A: We need to determine if the given subsets of the vector spaces are subspace or not. This can be done…

Q: Given two n xn matrices A and B, prove that the set S = {x € R" : ABx = 0} is a subspace of R".

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

Q: Which of the following sets are subspace of M2x2? (a) All 2 x 2 matrices of the form ( 2) where…

A: We find which of the given sets are subspace of M2×2.

Q: Let V be a vector space over R, and let Sị and S2 be subspaces of V. (a) Prove that Si C S2 =…

A: I have proved the all statements by taking a basis of vector spaces

Q: Let V be a vector space and let S = {v,,…, Vn} be a set of vectors in V. Prove ... that the span of…

Q: Which of the following sets are vector subspaces of R3 and why? i. S1 ={x in R3: x1 + x2 = 0, x1 +…

A: To determine the set S is a subspace if w1 , w2∈S then c1w1+c2w2∈S (i) Consider…

Q: Which of the following subsets of R" with the usual operations of vector addition and scalar…

Q: 2. Let V be an inner product space and v1, · · , Vn E V. For each of the following statements, •…

Q: 2. Determine whether W is a subspace of V = C³ where W = {(0,b, c) E C³}. %3D

Q: Exercise 1: For each of the following, determine whether it is a subspace of the given vector space,…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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: Suppose W1 and W2 are subspaces of a vector space, V. Define W as follows: W = {w1 + W2 | W1 E W1…

A: As W1 is a subspace of V , hence (a) 0∈W1 (b) αw1+βw1'∈W1 for every w1 , w1'∈W1 As W2 is a subspace…

Q: Consider the vector space of real 3 x 3 matrices M3,3. i: Consider S {A € M3,3|A = A"} Show that S…

A: Since you have asked multiple questions, we will solve first one for you. If you want other question…

Q: Let A and B be m x n matrices. Show that V = {x e R" : Ax = Bx} is a subspace of R".

Q: Which of the following subset is the subspace of the specified vector space? I. T = {(x,0, z,0) |…

Q: Consider the vector space of 2 x 2 matrices with the usual addition and scalar multiplication of…

Q: Let W1,W2,K1,K2, . . . ,Kp,M1,M2, . . . ,Mq be subspaces of a vector space V such that W1 = K1⊕K2⊕·…

Q: Let the set R(2×2) of 2×2-matrices be a vector space. Let A∈R2×2, (I) be the identity matrix, and…

Q: Which of the following is a subspace of the vector space P2 of all polynomials of degree at most 2?…

Q: Let W, and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Given that W1 and W2 are the sub spaces of a finite dimensional vector space V.

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

Determine which of the following sets are subspaces.

W1={[acbd]∈R2×2∣ad−bc≥0}

W2={(x,y,z,t)∈R4∣2x+3y=z−t}

Justify your answer !!!

Please be reminded that R2×2 is the vector space of 2×2 matrices, and, R4 is the vector space of all 4-tuples…

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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 with 1 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, algebra and related others by exploring similar questions and additional content below.

Similar questions

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.
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?
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.
In Exercises 1-4, let S be the collection of vectors in [xy]in2 that satisfy the given property. In each case either prove that S forms a subspace of 2 or give a counterexample to show that it does not. xy0
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.
Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[a0a]}