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 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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 =
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