10. Let S be the subspace of P2(R) consisting of all polynomials p such that p(1) = 0. Find a basis for S. What is the dimension of S?

Q: 1. Consider the following subspaces of R³: U = {(x,x,y): x,y ER} and W = {(x,-x, 2x): x, y ≤ R} (a)…

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Q: Find the orthogonal projection of 3 onto the subspace W of R* spanned by 1 -1 -1 projw (7) = %3|

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Q: 4. Assume that T E L(V) and that U1 and U2 are T-invariant subspaces of V. Prove that U1 n U2 is…

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Q: 142 Q2. Let A = Determine a basis for each of the subspaces CS(A"), |3| 316 N(A), CS(A), and N(A").

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Q: 5.7.5 ND Suppose that V has an inner product, and that n is in V. Show that the set Vo of all…

A: We will use the definition to prove that, V0 is a subspace of V, keeping in mind the properties of…

Q: 2. Find a basis and the dimension of the subspace W = {(0,a+b,a-b); a,be R}

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Q: -10 e subspace W of R" spanned by 6 and 3

A: In this question, concept of orthogonal projection is applied. Orthogonal Projection Matrix of…

Q: 6. For each of the following subspaces W of R³, find the standard matrix of the linear trans-…

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Q: 5. Let W be the set of all vectors p(t) in P2 such that p(0) = 0. Show that W is a subspace of P,…

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Q: 2.14. Prove that two subspaces V1 and V2 are linearly independent if and only if Vịn V2 = {0}.

A: V1 and V2 are two subspace.

Q: Z. Suppose A and B are m x n matrices. Prove that C(A) and C(B) are orthogonal subspaces of R" if…

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Q: Find a basis for the subspace of R3 spanned by S.S = {(1, 2, 4), (−1, 3, 4), (2, 3, 1)}

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Q: Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}

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Q: Find the orthogonal projection of v = projv (v) = [B] 9 onto the subspace V of R³ spanned by 4 and…

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Q: 7 4. 3 and let W the subspace of IR spanned by u and v. Find a basis of W, the orthogonal complement…

A: The objective is to find orthogonal complement of W

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1– t°} and G= Span{1+t+t°, t – t°, 1+t+t}…

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Q: Determine whether the following are subspaces of C[−1, 1]: The set of odd functions in C[−1, 1]

A: Consider the given set: C[−1, 1] Here the objective is to prove that the set of odd functions in…

Q: Consider the following vector v ER and the subspace SCR¹. -0-000 S= = span V= V₂ (a) Find an…

A: consider the following vector v∈ℝ4 and the subspace S⊂ℝ4 where v=1234 and S=span1111, 01-11, 0110…

Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1-t3 } and G = Span{1+t+t 2, t – t 3, 1+t+t…

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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: 1 2 Find a basis for subspace V of R4 spanned by the vectors 3 dimension of V? 8 What is the

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Q: 8. Q4) Let 5 } is a basis for a subspace W. Use the Gram-Schmidt prosses to produce an orthogonal…

A: Here, let v1=30-1, v2=85-6 are the basis element for a subspace of W.

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1 –} and G Span{1+t + , t-, 1+t+*} Find dim…

A: According to the given information. Consider the subspaces of P as:

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1– t°} and G=Span{1+t+t°, t – t°, 1+ t+t®}…

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Q: (3) Suppose that V is a finite dimensional vector space over R and T € L(V) has no eigenvalues.…

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Q: (1) Suppose U and W are both five-dimensional subspaces of R°. Prove that UnW + {0}.

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Q: 4. Let S be the subspace of P₂ defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim (S), and show…

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Q: 3.12. Show that for a subspace E we have (E-)- = E. Hint: It is easy to see that E is orthogonal to…

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Q: Find the orthogonal projection of -5 -1 16 onto the subspace W of IR³ spanned by 4 -4 and 2 24 projw…

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Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠S2 and S2≠R4.

Q: 4. Let be B = {1,eª, xe²} a basis of a subspace E of the space of continuous functions, and let D.…

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Q: 2. Consider the following vector v R¹ and the subspace SCR¹. S = span (a) Find an orthonormal basis…

A: Note:- As per our guidelines, we can answer first part of this problem as exactly one is not…

Q: 2. Show that {1,x – 1, x (x + 1)} is a basis of the space of polynomials P2 (IR) and that W = {p(x)…

A: If the elements of the set are linearly independent then the set is a basis set. If Wvis a subspace,…

Q: Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive…

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Q: 3.) Determine the dimension of the subspace of P, consisting of all polynomials of the fom…

A: Dimension of subspace of p2 is 3 .

Q: Given a basis B = {V1, V2,, Vn} of Subspace S, there is a unique way to express any ... vE S as a s…

A: We will first write 2 expressions of v as a linear combination of basis vectors in B. Then, we will…

Q: Find a basis for the subspace of R³ spanned by S. S= {(4,2,-1), (1,2,8), (0,1,2)}

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Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

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Q: (2) Find a basis for the subspace W%3D { (а+c, 2a — b, За — 2b — с, b + 2с) | а, b, с € R} of R4.

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Q: 5. Consider the vector space P2(R), the set of all polynomial func- tions of degree less than or…

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Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: 4. Let S be the subspace of P2 defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim(S), and show how…

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Q: Find the orthogonal projection of -13] 15 onto the subspace W of R4 spanned by 4 and 5 -2 -2…

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Q: Find the orthogonal projection of [-10] = -15 12 onto the subspace W of IR spanned by 日-日 1 9. and 6…

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Q: (c) Let M = (xn)el² :x, + x, = 0}. Show that M is a closed subspace and find an orthonormal basis…

A: Please check step 2 for the solution.!

Q: 2 Find the dimension of the subspace H of R spanned by – 12 8 ..... dim H =

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Q: (5) Suppose that T E L(V) is such that all subspaces of V of dimension dim(V) – 1 are T-invariant.…

A: Please check step 2 for the solution.!

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.