Suppose the V=(V,+,*) is a vector space over the scalar field of real numbers with scalar...
Suppose the V=(V,+,*) is a vector space over the scalar field of real numbers with scalar multiplication * and vector addition +; and suppose V has Dimension 8. If W1 is a subspace of Dimension 5 and W2 is a subspace Dimension 4, then what are the possible dimensions of the subspace W1nW2? (The intersection of W1 and W2)
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Suppose the V=(V,+,*) is a vector space over the
scalar field of real numbers with scalar...
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V...
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V of dimension seven. Explain why W1 n W3 {0 (b) Suppose V is a vector space of dimension 55, and let Wi and W2 be subspaces of V of dimension 36 and 28 respectively. What is the least possible value and the greatest possible value of dim(Wi + W2)?
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real...
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real valued functions on on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W-f eV: f(0)-0is a subspace of V or not? (b) Verify, if the set W-Uev f(0) 1 is a subspace of V or not? (c) Verify, if the set W-İfEV:f(x)-0V-2-z is a subspace of V or not? 1b) PrtScn Home FS F6 F7 F8 5
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº =...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
Determine if the set V = {at? | a € R} is a subspace of the...
Determine if the set V = {at? | a € R} is a subspace of the vector space P2 = {ao +ajt + azt? | ao, a1, az ER}. You may assume that vector addition in P2 is given by the usual addition of polynomials and that the scalars used in scalar multiplication are real numbers. If you decide that Vis a subspace of P2, then identify the zero vector in V and explain briefly why Vis closed under vector...
6. (i) Prove that if V is a vector space over a field F and E...
6. (i) Prove that if V is a vector space over a field F and E is a subfield of F then V is a vector space over E with the scalar multiplication on V restricted to scalars from E. (ii) Denote by N, the set of all positive integers, i.e., N= {1, 2, 3, ...}. Prove that span of vectors N in the vector space S over the field R from problem 4, which we denote by spanr N,...
Let V = M2x2 be the vector space of 2 x 2 matrices with real number...
Let V = M2x2 be the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication. Which of the following subsets form a subspace of V? The subset of upper triangular matrices. The subset of all matrices 0b The subset of invertible matrices. The subset of symmetric matrices. Question 6 The set S = {V1, V2,v;} where vi = (-1,1,1), v2 = (1,-1,1), V3 = (1,1,-1) is a basis for R3. The vector w...
just part c,d, and e please!! Let V be a finite-dimensional vector space over F. For...
just part c,d, and e please!!
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...
(f) (4 points) The set of ordered pairs of real numbers V= | 1VER is a...
(f) (4 points) The set of ordered pairs of real numbers V= | 1VER is a vector space with vector addition and scalar multiplication defined by [*1 + x2 - 1] Lyı + y2 - 1 k(y - 1)+1] Verify that there exists a vector ✓ EV such that ✓ D7=@ ū = 7, for any ve V.
1. Why the following sets are not vector space? with the regular vector addition and scalar...
1. Why the following sets are not vector space? with the regular vector addition and scalar multiplication. a) V = {E: * > 0, y 20 with the regula b) V = {l*: *y 2 o} with the regular vector addition and scalar multiplication. c) V = {]: x2+y's 1} with the regular vector addition and scalar multiplication. 2. The set B = {1,1+t, t + t2 is a basis for P, the set of all polynomials with degree less...
Need to use all axioms to prove this is a vector space. e(a+b)z and scalar multiplication...
Need to use all axioms to prove
this is a vector space.
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a vector space under these definitions? If so, what is the 0 element = eaeba- 8. Let V = k ea of V?
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a...
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V of dimension seven. Explain why W1 n W3 {0 (b) Suppose V is a vector space of dimension 55, and let Wi and W2 be subspaces of V of dimension 36 and 28 respectively. What is the least possible value and the greatest possible value of dim(Wi + W2)?
2. On subspaces of C(-1,1) Let V C(-1,1) be the vector space of all continuous real valued functions on on the interval (-1, 1), with usual addition and scalar multiplication. (a) Verify, if the set W-f eV: f(0)-0is a subspace of V or not? (b) Verify, if the set W-Uev f(0) 1 is a subspace of V or not? (c) Verify, if the set W-İfEV:f(x)-0V-2-z is a subspace of V or not? 1b) PrtScn Home FS F6 F7 F8 5
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
Determine if the set V = {at? | a € R} is a subspace of the vector space P2 = {ao +ajt + azt? | ao, a1, az ER}. You may assume that vector addition in P2 is given by the usual addition of polynomials and that the scalars used in scalar multiplication are real numbers. If you decide that Vis a subspace of P2, then identify the zero vector in V and explain briefly why Vis closed under vector...
6. (i) Prove that if V is a vector space over a field F and E is a subfield of F then V is a vector space over E with the scalar multiplication on V restricted to scalars from E. (ii) Denote by N, the set of all positive integers, i.e., N= {1, 2, 3, ...}. Prove that span of vectors N in the vector space S over the field R from problem 4, which we denote by spanr N,...
Let V = M2x2 be the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication. Which of the following subsets form a subspace of V? The subset of upper triangular matrices. The subset of all matrices 0b The subset of invertible matrices. The subset of symmetric matrices. Question 6 The set S = {V1, V2,v;} where vi = (-1,1,1), v2 = (1,-1,1), V3 = (1,1,-1) is a basis for R3. The vector w...
just part c,d, and e please!!
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...
(f) (4 points) The set of ordered pairs of real numbers V= | 1VER is a vector space with vector addition and scalar multiplication defined by [*1 + x2 - 1] Lyı + y2 - 1 k(y - 1)+1] Verify that there exists a vector ✓ EV such that ✓ D7=@ ū = 7, for any ve V.
1. Why the following sets are not vector space? with the regular vector addition and scalar multiplication. a) V = {E: * > 0, y 20 with the regula b) V = {l*: *y 2 o} with the regular vector addition and scalar multiplication. c) V = {]: x2+y's 1} with the regular vector addition and scalar multiplication. 2. The set B = {1,1+t, t + t2 is a basis for P, the set of all polynomials with degree less...
Need to use all axioms to prove
this is a vector space.
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a vector space under these definitions? If so, what is the 0 element = eaeba- 8. Let V = k ea of V?
e(a+b)z and scalar multiplication as feax a E R} define addition as ea* + ebx ekax where k e R. Is V a...
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