Bravais basis lattice vectors b,, b2, b3 are defined with the real space basis vectors a1, a2, az in the following: 2π(a, x α,) ,b2 [а, (аz х аз)1" 2n (az x a,) 2π(α x ,) b1 and b3 [а, (аz х аз)] [a1 · (a2 x a3)] 27(b2xb3) [b1•(b2×b3)] Prove its inversion vector relation a, = etc by vector identities like A × (B × C) = В (A- С) — С(А В) and A - (В x С) B· (C × A) = C ·(A × B). %|

Let V₁, V2, V3 be the vectors in R³ defined by 18 ---D V₁ = -6 V2 = -14 V3 = 20 (a) is (V1, V2, Vs} linearly independent? Write all zeros if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of V₁, V₂, and V3 0 (c) Type the dimension of span {V1, V2, V3}: Note: You can earn partial credit on this problem. -25 -18] 0=v₁+√₂+vs (b) Is (v1, vs} linearly Independent? Write all zeros if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of V₁ and V3. 0=v₁+v₁ V3

Prove that the vectors (?\index{1},?\index{2}), (?\index{1},?\index{2})∈R×R are linearly dependent iff ?\index{1}?\index{2}-?\index{2}?\index{1}=0.

Make the vectors u1=(2,0,-1,1), u2=(1,1,-1,3), u3= (-1,1,1,-1) orthonormal.

Let U be a subspace of R", and let S = { ū, v, w } be a set of three non-zero vectors in R”. Fill in the blanks so that the following statements are true. If you believe you don't have enough information to conclude anything for any of the cases, choose the answer "???". Note that there is no intended relationship between the parts. In each case, assume only what the statement says to assume. If SCU, then dim(U) > (Enter the largest integer that makes the statement necessarily true.) If S is linearly independent and S C U, then dim(U) 3. If S' is a spanning set for U, then dim(U) 3. If span(S) = U then S If S is a basis for U, then S linearly independent. linearly independent.

Q1// Determine āxb if ā=3i-5 j+7k and b=2i-6j+4k and check if resultant vector are perpendicular to à and

Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(2, –1, 3), (5, 0, 4)} (a) z = (3, –4, 8) - - ( 4 )a, • ( 1 Z = (») v-(10, -) 1 59 V = 18, 4 1 7 )e, + ( V = 4 (c) w = (3, -9, 15) ( 9 s, + ( -3 W = (d) u = (4, 1, –1) u =

Solve a part only in 20 minutes plz i need it plz

3) Let C be a basis that contains vectors 2 (−2 + 1x + 1x^ + (−1) x3), (1 + (−2) x + 4x2 + 2x3), (−2 + 1x + 1x2 + (-1) 3), (−4 + 4x + 2 + 4x3) and P = C-B -1 Then : b₁ b2 = , b3 : = , ba = 3 = 4 4 -1 4 -24 -2 -2 2 ↓ NALN -1 -4 -2

Let V1, V2, V3 be vectors in R³ given by 1 3 V₁ = -- ()⋅ ⋅=()· ~= (²³). 6 V2 2 V3 5 2 0 The set {V1, V2, V3} is linearly dependent. Find scalars C1, C2, C3, not all zero, such that CIVI + C₂V2 + C₂V3 = 0.