Questions and Answers of Linear Algebra

Find the matrix [T]C†Bof the linear transformation T : V †’ W with respect to the bases B and C of V and W, respectively. Verify Theorem 6.26 for the vector v by computing T(v)
Test the sets of functions for linear independence in F. For those that are linearly dependent, express one of the functions as a linear combination of the others.{1, ln(2x), ln(x2)}
Find either the nullity or the rank of T and then use the Rank Theorem to find the other.T : P2 → R defined by T(p)x)) = p′(0)
Test the sets of functions for linear independence in F. For those that are linearly dependent, express one of the functions as a linear combination of the others.{sin x, sin 2x, sin 3x}
Let T : R3†’ P2be a linear transformation for whichand Findand тH 3D 1 — 2х 3 = x + 2x² -1
Determine whether the linear transformation T is (a) one-to-one and (b) onto.T : P2 †’ M22 defined by a + 2c a + b T(a + bx + cx²) = 2a + c
Determine whether the set B is a basis for the vector space V. V = M2, B %3D {P JE9}
Determine whether the set B is a basis for the vector space V. V = M2, B =
If T : U → V and S : V → W are linear transformations such that range(T) ⊆ ker(S), what can be deduced about S ͦ T?
Determine whether the set B is a basis for the vector space V. -1 V = M, B = 22 1 -1
Prove Theorem 6.33(b).Let S be the solution space ofy'' + ay' + by = 0and let λ1 and λ2 be the roots of the characteristic equation λ2 + aλ + b = 0.b. If λ1 = λ2, {eλ1t, eλ2t}then is a
Determine whether the set B is a basis for the vector space V. Myi8 = {: 1 2 2 2 2 3 V = M2»B 2'L- 3 2 2 1
Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space over the indicated Zp. If it is not, list all of the axioms that fail
Prove Theorem 6.14(b).Let T : V→W be a linear transformation. Then:T(-v) = - T(v) for all v in V.
Determine whether the set B is a basis for the vector space V.V = P2, B = {x, 1 - x, 1 + x + x2}
Prove Theorem 6.1(a).Let V be a vector space, u a vector in V, and c a scalar.a. 0u = 0b. c0 = 0c. (-1)u = -ud. If cu = 0, then c = 0 or u = 0.
Which of the codes are linear codes? {8-8-8} |C =
Determine whether the set B is a basis for the vector space V.V = P2, B = {1 - x, 1 - x2, x - x2}
Prove Theorem 6.1(c).Let V be a vector space, u a vector in V, and c a scalar.a. 0u 0b. c0 0c. (-1)u-ud. If cu 0, then c 0 or u 0.
Which of the codes are linear codes? {B.A} C =
Determine whether the set B is a basis for the vector space V.V = P2, B = {2 + 3x - x2, 1 + 2x + 2x2}
Determine whether V and W are isomorphic. If they are, give an explicit isomorphism T : V → W. V = P2, W = {p{x} in P3 : p(0) = 0}
Is M22 spanned by 1 1 1 1
Is M22spanned by -1 1 11
a) Prove that if a transposition error is made in the fourth and fifth entries of the ISBN-10 [0, 6, 7, 9, 7, 6, 2, 9, 0, 6], the error will be detected.b) Prove that if a transposition error is made
Find the distance between the parallel lines. -2 and 3 -2 х 3 ||
Write the equation of the line passing through P with normal vector n in (a) normal form and (b) general form. 3 P = (2, 1), n = -4

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