For Problem #12, how do I prove that the set is a basis for V? I think that infinity is the basis, but I'm not sure. This is a Linear Algebra type of question. Here is a picture. ing techniques from Chapter 5, we can show that det(C) = -59. Since theminant of the coefficient matrix C is nonzero, it follows from Theorem 5.7 (the Bigrem, Version 7) in Section 5.2 that our linear system has only the trivial solution.efore V is a linearly independent set and is a basis for P.DAVID Ithe set V could pos-10,VRx211. V- ((1,0, 0, 0, 0, ...), (1,-1, 0, 0, 0, ...)(1,-1, 1, 0,0,.), (1,-1, 1,-1,0,...),. ., VR12. (1, x+ 1, x+x+1, x+ x2+ x+ 1. ). V=PV PVR2For Exercises 13-18, find a basis for the subspace S and determinedim(S).13. S is the subspace of R consisting of matrices with traceequal to zero, (The trace is the sum of the diagonal terms of amatrix.)VRx214. S is the subspace of P2 consisting of polynomials with graphscrossing the origin.15. S is the subspace of R2x2 consisting of matrices with compo-basis for V.nents that add to zero.16. S is the subspace of T(2, 2) consisting of linear transforma-tions T: R R2 such that T (x) = ax for some scalar a.+ 1, V= P17. S is the subspace of T(2, 2) consisting of linear transforma-tions T: RVRx2R such that T(v) = 0 for a specific vector v.

Answered: ing techniques from Chapter 5, we can…

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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For Problem #12, how do I prove that the set is a basis for V? I think that infinity is the basis, but I'm not sure. This is a Linear Algebra type of question. Here is a picture.

ing techniques from Chapter 5, we can show that det(C) = -59. Since the
minant of the coefficient matrix C is nonzero, it follows from Theorem 5.7 (the Big
rem, Version 7) in Section 5.2 that our linear system has only the trivial solution.
efore V is a linearly independent set and is a basis for P.
DAVID I
the set V could pos-
10,
VRx2
11. V- ((1,0, 0, 0, 0, ...), (1,-1, 0, 0, 0, ...)
(1,-1, 1, 0,0,.), (1,-1, 1,-1,0,...),. ., VR
12. (1, x+ 1, x+x+1, x+ x2+ x+ 1. ). V=P
V P
VR2
For Exercises 13-18, find a basis for the subspace S and determine
dim(S).
13. S is the subspace of R consisting of matrices with trace
equal to zero, (The trace is the sum of the diagonal terms of a
matrix.)
VRx2
14. S is the subspace of P2 consisting of polynomials with graphs
crossing the origin.
15. S is the subspace of R2x2 consisting of matrices with compo-
basis for V.
nents that add to zero.
16. S is the subspace of T(2, 2) consisting of linear transforma-
tions T: R R2 such that T (x) = ax for some scalar a.
+ 1, V= P
17. S is the subspace of T(2, 2) consisting of linear transforma-
tions T: R
VRx2
R such that T(v) = 0 for a specific vector v.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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