Verify that the set in Example 6 is a subspace. Find a basis for thissubspace. Is {x2 + x + 1, x + 5, 3} a basis? I EXAMPLE 6 The set {a,x? + a,x + a, I a, a, a, E R} is a subspace ofthe vector space of all polynomials with real coefficients over R.

Answered: Image /qna-images/answer/ac72212b-c3b9-4d8b-9acc-cab7ba33a2af.jpg

Answered: I EXAMPLE 6 The set {a,x? + a,x + a, I…

I EXAMPLE 6 The set {a,x? + a,x + a, I a, a, a, E R} is a subspace of the vector space of all polynomials with real coefficients over 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: Let V be the vector space of symmetric 2 x 2 matrices and W be the subspace -5 = span{ 2 -3 -3 -1 a.…

Q: 2. Let S be the subspace of R5 spanned by x = (1, , (a) Find a basis for S- (b) Give a geometrical…

Q: Is the set of all vectors (x, y) in R, with the usual addition and scalar multiplication, a subspace…

Q: a 2. Is the set of vectors of the form a subspace of R? Show your proof

A: **As per our guidelines we are allowed to answer only one question at a time. Kindly please re-post…

Q: a. Let V be any finite dimensional vector space. In class, we learned that for any finite set (w1,…

A: To Check: Whether the given statement is true or false. Statement: If W is a subspace of a vector…

Q: Let V be the vector space of symmetric 2 × 2 matrices and W be the subspace -3 W = span{ 1 1 }. -3…

A: In this question, concept of span is applied. Linear Span Let V represent a vector space, and…

Q: A subspace U of R^2 is a subset which satis es three properties: A) 0∈U, B) U is closed under…

Q: a Show that the set of vectors of the form a +b forms a subspace of R³. Find a 2а - b basis for this…

A: A set S is said to be a subspace of a vector space V if it satisfies the following properties: For…

Q: 2. Let H be the set of all vectors of the form (a – 3b, b – a, a, b) where a and b are arbitrary…

A: Let V be a vector space. H is said to be a subspace of V. If H is a subset of V. H is closed with…

Q: Let V be the vector space of symmetric 2 x 2 matrices and W be the subspace 3 -5 W span{ -4 -5 2 a.…

Q: ) 3) Let {rn :n e N} be a basis in a normed space X. Show that X is not a Banach space. (Hint: Apply…

Q: Theorem 6.8. Let A be a closed subspace of a compact space. Then A is compact.

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

A: The dimension of a nonzero subspace H, denoted by dim(H), is the number of vectors in basis for H.

Q: snip

A: Let x1=a1b1-b1a1 and x2=a2b2-b2a2 Then,…

Q: 1. Show that WW10 W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: 1. Let W = {( 2).a+ 2d = 0}. Show that W is a subspace of the set of all 2 x 2 matrices and find its…

Q: 4. Let S be the subspace of R4 spanned by x = (1,0,-2,1)7 and x2 = (0,1,3, -2)T. Find a basis for S

A: Consider a matrix A as follows.

Q: 6. (a) Determine whether the set of all vectors of the form (a,b,c), where b a+c, is a subspace of…

A: The subspace of R3 is (a,b,c) where b=a+c

Q: Verify that the set <v1, v2, . . . , vn> defined in Example 7 is a subspace.

Q: 2. Let V = E3 and W = {x|x = a(1,0,2) + b(1,-1,3)} and S = {x\x = (x1,x2, 1)} where a and b are any…

Q: 3. Let A be an arbitrary m x n matrix. Consider all x's, such that Ax = 0, and show that the set of…

Q: Theorem 6.9. Let A be a compact subspace of a Hausdorff space X. Then A is closed.

Q: 3. This problem is about set addition and subspaces. Let d = and dz = and V = span{d}, %3D U =…

Q: Use Theorem 4.2.1 to determine whether the set of all vectors of the form (0,0,m) form a subspace of…

A: Option first is correct

Q: a Show that the set of vectors of the form a +b forms a subspace of R³. Find a 2а — b basis for this…

A: Consider two vectors of the given form a=a1,a1+b1,2a1+b1 and b= a2,a2+b2,2a2+b2 Let α∈ℝ Then…

Q: Give an example of a vector space V and non-trivial subspaces X.Y. Z of V such thatV =X@Y= X@Z but Y…

A: Give an example of a vector space V and non-trivial subspaces X, Y, Z of V such that…

Q: A. W = {(x1,0,1)| x, is a real number} under the standard operations in R°is a subspace of R³. B. W…

Q: how that if T is a linear operator on V and W1, W2 are T-invariant subspaces of V, en W1+ W2 is a…

Q: Prove that if V is a vector space and S = {v1,v2,…,,vp } is any given set of vectors in V, then…

Q: A) The set of all ax³ - bx² - Select one: O True O False polynomials of the form cx + 2, where a,b…

Q: (a) Determine whether the set of all vectors of the form (a,b,c), where b= a+c, is a subspace of R³.…

A: Consider the subset S of ℝ3 defined as, S=(a,b,c)∈ℝ3 | b=a+c Take a1,a2,a3,b1,b2,b3…

Q: Determine whether the subset of C(-0, co) is a subspace of C(-0, co) with the standard operations.…

Q: 3. If A and B are finite-dimensional, orthogonal subspaces of an inner product space V, then show…

Q: 3. This problem is about set addition and subspaces. Let d and d2 , and V = span{di}, U = span{d2}.…

Q: 6 Give a counter example to show that W is not subspace of R3. W=set of (x, y, z) where x + y = 1

Q: a) Determine whether the set of the invert 2 X 2 matrix is a subspace of the vector space Max2?…

Q: A) The set of all polynomials of the form ax³ - bx² - cx + 2, where a,b and can real numbers is a…

A: Let, S be the set containing all the polynomials of the given form. i.e., S = {ax3 - bx2 - cx + 2 :…

Q: Consider the subspaces U = span{[1 -4 -5],[-1 –6 -13]} and W = span{[4 -4 4],[-2 0 -2]} of V = R'x3.…

A: Any vector belongs to a space spanned by v1 and v2 is the linear combination of v1 and v2.

Q: Consider the subspaces U = span{[3 -3 1],[12 3 -7]} and W = span{[-3 2 0],[1 8 -6]} of V = R1x3.…

Q: Use subspace criteria to show that W= the set of all upper triangular matrices form a subspace of…

A: Use subspace criteria to show that W= the set of all upper triangular matrices form a subspace of…

Q: Consider the subspaces U = span{[3 5 0],[0 12 -6]} and W = span{[–3 2 -1],[-15 7 -11]} of V = RIX3.…

Q: 3. Let W be a subspace of R" of dimension k < n. Use the Fundamental Theorem of Linear Algebra to…

Q: Let 9. 2 H = span 10 A basis for the subspace H C R* is { }. vector

Q: Use Theorem 4.2.1 to determine whether the set of all vectors of the form (a,0,0) form a subspace of…

Q: 1. Determine whether the set W =- ER,x, is a subspace of R' with the standard operations. Justify…

A: Given the set W=x1, 1x1, x3 | x1, x3∈R, x1≠0 We have to determine whether W is a subspace of R3with…

Q: Please solve this question with a good explanation. It is Linear Algebra question.

A: Let W be the set of traceless matrices.

Q: 1. Give examples of subspaces of " and 1 which are not closed. 2. What is the largest possible c in…

A: Multiple : Consider , C00 the collection of the sequence with finite support. That is , for each…

Q: 1. Use the definition of a subspace to prove or disprove: W is a subspace of M, (you must verify all…

A: Given that W is the set of all 3×2 matrices of the form a+4b00ba3b.

Q: 6. (a) Determine whether the set of all vectors of the form (a,b,c), where b = a+c, is a subspace of…

A: To show the subset W is a subspace , we need to show that (a) 0∈W (b) For every w1 , w2∈W , we have…

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

Verify that the set in Example 6 is a subspace. Find a basis for this
subspace. Is {x2 + x + 1, x + 5, 3} a basis?

I EXAMPLE 6 The set {a,x? + a,x + a, I a, a, a, E R} is a subspace of
the vector space of all polynomials with real coefficients over R.

Expert Solution

Step 1

Step 2

Step by step

Solved in 3 steps with 3 images

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

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.
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.
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are nonnegative.
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. 37. V = P, W is the set of all polynomials of degree 3
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose second component is the square of the first.
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are Pythagorean triples.
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all matrices in Mn,n with determinants equal to 1.
Subsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose components are integers.
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?