1. Consider the following two bases for R3. --{():(1) 0)) -= {(8) 0) (1) and (a)...
15 points) Consider the following vectors in R3 0 0 2 V1 = 1 ; V2 = 3 ; V3 = 1] ; V4 = -1;V5 = 4 1 2 3 = a) Are V1, V2, V3, V4, V5 linearly independent? Explain. b) Let H (V1, V2, V3, V4, V5) be a 3 x 5 matrix, find (i) a basis of N(H) (ii) a basis of R(H) (iii) a basis of C(H) (iv) the rank of H (v) the nullity...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
2. Consider the set 8- ( B = ) in R3 (a) Show that B is a basis for R3. Be sure to justify your answer. (b) If v = ei + €2, compute (v]. (c) Consider the basis =([A] AM) (you do not have to justify that this is a basis). Compute (1)B' and (v]B, where v is the vector from part (b).
Consider the bases B = {U1, U2} and B' = {u', u'z} for R2, where 6 1 u = u2 = U2 = -1 -1 2. 5 Compute the coordinate vector [w]B, where W = [3 7 3 and use Formula (12) [v]s' = P. PB-8 [V]B ) to compute [w]g' [w]B = ? Edit [w] II ? Edit
(i) Find an orthonormal basis {~u1, ~u2} for S
(ii) Consider the function f : R3 -> R3 that to each vector ~v
assigns the vector of S given by
f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a
linear function.
(iii) What is the matrix of f in the standard basis of R3?
(iv) What are the null space and the column space of the matrix
that you computed in the
previous point?
Exercise 1. In...
Chapter 4, Section 4.6, Question 03b Consider the bases B = {u, u, uz) and B' - {u', u', u'3) for R3, where 2 1 2 -1 3 u = U2 = [i uz = 2 1 u -13) u2 1 1 -3 из 2 Compute the coordinate vector (w]g, where w = | [-71 -4 and use Formula (12) [v]B = P8-8 [v]B ) to compute [w |-7 [w] = ? Edit [w] B 11 Edit Chapter 4, Section...
o 1 0 -1 Exercise 2. Let A= in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R3, set g(W) = WT AT ER. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]C,B in the - 1 bases C = {1} and B { 8.00 } ? (ii) Let f : R3 → R be the function defined by f() = 7T Aw E R. Show that...
2. The following are ordered) bases for a subspace S of R3. (a) (8 pts. Find the change-of-basis matrix from B, to B.
Determine which sets in Exercises 1-8 are bases for R3. Of the sets that are not bases, determine which ones are linearly independent and which ones span R. Justify your answers. ( 0
2. Consider the polynomials 0-k (z) := (1 + z) for k-0,..., 10 and let B-bo,b1bo) can be shown that B is a basis for Pio the vector space of polynomials of degree at most 10. (You do not need to prove this.) Let Pk (z)-rk for k = 0, 1, . . . , 10, so that S = {po, pi, . . . , pio) is the standard basis for P10. Use Mathematica to: (a) Compute the change...