5. Find the dimension of the subspace of P, spanned by the given set of vectors: (a) {r2, r? +1, x² + x}; (b) {r? – 1, x + 1, 2r + 1, r2 – a}. 6. Indicate whether the following statements are always true or sometimes false. For true statements, give a proof, and for false statements, give a counter-example. (a) If {v1, v2} is a linearly independent set of vectors, then {2v2, v1 + v2} is linearly independent. (b) Let S and T be linear maps V→ W. Let u, v, w e V such that S(u) = T(u), S(v) = T(v), and S(w) = T (w). Then S(a) = T(x) for all a E span{u, v, w}. %3D {() 0 0 7. (Consider the basis B = 3 of R3. Find the coordinate vector of v with 4. respect to B, where: (a) v =

5. Find the dimension of the subspace of P, spanned by the given set of vectors: (a) {r2, r? +1, x² + x}; (b) {r? – 1, x + 1, 2r + 1, r2 – a}. 6. Indicate whether the following statements are always true or sometimes false. For true statements, give a proof, and for false statements, give a counter-example. (a) If {v1, v2} is a linearly independent set of vectors, then {2v2, v1 + v2} is linearly independent. (b) Let S and T be linear maps V→ W. Let u, v, w e V such that S(u) = T(u), S(v) = T(v), and S(w) = T (w). Then S(a) = T(x) for all a E span{u, v, w}. %3D {() 0 0 7. (Consider the basis B = 3 of R3. Find the coordinate vector of v with 4. respect to B, where: (a) v =

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 67E: Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many...

See similar textbooks

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=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=Mnn,WAinMnn:detA=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=3, W={[aba+b+1]}
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
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.
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?
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.