Jump to level 1 Let (U₁=[2₁2] ₂-[-1818)-[3]} U3 = 0 Use the Gram-Schmidt process to find an orthogonal basis under the Frobenius inner product. Orthogonal basis: V₁ a = Ex: 5 b: 0 0 a 1.59 * {v₁ = [²₁₂2] ·M = [8 -¹8]. ³₁ = [1185 2]} V₂ V3 -1 -1.65 Ex: 5 0 C = Ex: 1.23 be a basis for a subspace of R2x2. d=Ex: 1.23

[ 5/4 -3/4] -3/4 5/4. Let A write A UAU-1 where U is unitary, columns are orthonormal basis for R? and A = with A1 > A2. Recall: U-1 = UT for unitary U.

Find w if (w)s= W = [] -11 1 ΓΟΤ relative to the basis S = {3 3 " 1 1 ' 3 }

(2, 4, 0), the Gram-Schmidt process Given vị = (-1, 2, –1), v2 = (1, 7,1) and v3 allows us to construct an orthonormal basis {u1, u2, U3 } of R³ such that u¡ is a multiple of V1, u2 is a linear combination of vj and v2, and uz is a linear combination of v1, V2 and v3. Then u2 is equal to: O (1,1, 1) O (1,2, 0) (1:2'0뚜 이 O (0, 1, 0) o (-1, 2, –1) ㅇ 늘(1,0, 1)

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[4 -1 6] Given that 2 is an eigenvalue for A = |2 [2 1 Find a basis of its -1 8 Eigen space.

The set -23 (-0-0-0-0) 16 is a basis for R Use the Gram-Schmict process to produce an orthogonal basis for ¹. 111 = V₁ = U₂ = V₂ 11 VJ -15 -3 your answer" 11₁ = V₁ U₂ V₂. 11111 To make computations nice, if necessary, we can scale this vector by a non-zero scalar, so that y= ū₁ = us -U₁= 11V₂ HH 11₁ all To make computations nicer, if necessary, we can scale this vector by a non-zero scalar, so that u₂ = ₂ Va 1₂ U₂ u₂ V4. u₁u₂ 1₁ (Do not scale your answer -= -u₂ = 11₂ 11 (Do not scale r (Do not scale your answer) To make computations nicer, if necessary, we can scale this vector by a non-zero scalar, so that Therefore, [u, uz, us. u) is en orthogonal basis for R. To obtain an orthorormal basis, we scale each of these vectors as follows: Û₂ = u;= 3 1 113 VI 113 113 -1₂ = |u₁|| -1₂ = Note: To enter a number of the form type a/sqrt(n) Thus, (₁, ₂, 3, 4) is an orthonormal. basis for R 113= -6 -6 -15

Let + {U₁ = [₁ ], U₂₁ = [2₂8] · U₂ = [-2² , U3 20 Gram-Schmidt process to find an orthogonal basis under the Frobenius inner product. Orthogonal basis: V₁ a = Ex: 5 =[]} a 2 {v₁ = [1₂8], v₂ = [8²], b = Ex: 5 3], V₁ = [1 V3 c = Ex: 1.23 -1.33 1.33 be a basis for a subspace of R2x2. Use the d= a]} Ex: 1.23

Find a basis for p.

1 3. (а) Find an orthonormal basis B {ū1, ū2, đ3} of R³ such that u and the x-component of ūz is 1/3. [1 Find the coordinate vector of 3 relative to the basis B. (b) 1