3. (a)Show that B = {1+ 2t, t - t2,t +t} is a basis for P3.(b)) Let p(t) = 1+3t +t?. Compute [p(t)]s-4. In (a)-(d), A and B are two n xn matrices. State whether the statements below are true or false.Justify your answers.det(A" B) = det Adet B.S = {a+bt² : a,b E R} is a subspace of P3.) If rank(A) = n then Ar = 0 has a unique solution.(a)(d)If the columns of an n x n matrix A are linearly independent then the rows of A arelinearly independent.Let A be a 3 x 5 matrix and let rank(A) = 3. Then Ar = b has infinitely manysolutions for any b E R°.(e)5. Let U and V be two subspaces of R". W = U +V stands for a set of vectors of the form w = u+vfor some u e U and v e V.(a)Show that W is a subspace of R".(b)If UnV = {0} then dim W = dim U + dim V.

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Show that B = {1+ 2t, t – t2,t+t2} is a basis for P3. ) Let p(t) = 1+3t +t. Compute [p(t)]s- (a) (b)

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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Question

3. (a)
Show that B = {1+ 2t, t - t2,t +t} is a basis for P3.
(b)
) Let p(t) = 1+3t +t?. Compute [p(t)]s-
4. In (a)-(d), A and B are two n xn matrices. State whether the statements below are true or false.
Justify your answers.
det(A" B) = det Adet B.
S = {a+bt² : a,b E R} is a subspace of P3.
) If rank(A) = n then Ar = 0 has a unique solution.
(a)
(d)
If the columns of an n x n matrix A are linearly independent then the rows of A are
linearly independent.
Let A be a 3 x 5 matrix and let rank(A) = 3. Then Ar = b has infinitely many
solutions for any b E R°.
(e)
5. Let U and V be two subspaces of R". W = U +V stands for a set of vectors of the form w = u+v
for some u e U and v e V.
(a)
Show that W is a subspace of R".
(b)
If UnV = {0} then dim W = dim U + dim V.

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