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)
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb2ae333-5e69-44be-b97b-f2e5d9990b05%2F77d7cdc8-d9f0-4d4e-95d2-312342833570%2F9028mz_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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