--- Let S = span {x¹ = (a) Use Gram-Schmidt process to find an orthogonal basis {u¹, u²} of S. (b) Find an orthonormal basis {q¹ q²} of S. (c) Find a unit vector q³ such that Q = [q¹ q² q³] is a 3 × 3 orthogonal matrix.

The vectors v1, v2 and v3 form an orthonormal basis for R3 wherev1:=vector([0,0,-1]);v2:=vector([1/sqrt(2),1/sqrt(2),0]);v3:=vector([-1/sqrt(2),1/sqrt(2),0]);The coordinates of the vector w:=vector([1,-1,1]) with respect to the basis vectors v1, v2 and v3 of R3 are:

Show linear dependence in part (ii) to show that ({1,0,1}) is a basis.

Apply the Gram-Schmidt process to find an orthogonal basis for S. 2 pan{ [-²]. [8]} eBook S = span

V = Span ({··]}) Find an orthonormal basis of R³ that contains the vector 圓

Apply the Gram-Schmidt process to transform the basis vectors V1= (1,1,1), V2 = (0,1,1), V3 = (0,0,1) into an orthogonal basis (u1,u2,u3) and then normalize the orthogonal basis vectors to obtain an orthonormal basis (w1,w2,w3)

What is the basic way to find a basis for all of the vectors x y z  that satisfies the matrix equation of a 3 x 3 matrix

Let W = Lin in (1)-(²)) and U-Lim (2) U= (a) Show that R³ = WOU. Explain. (b) Find matrix P that represents a projection of R³ on W parallel to U. (c) Find vector u EU that will be closest to the vector z = 1 (d) Find an orthonormal basis of W and extend it to the orthonormal basis of R³.

The vectors V₁ basis for W. = 1 - 8 3 and V2 1 - 1 -3 form an orthogonal basis for W. Find an orthonormal The orthonormal basis of the subspace spanned by the vectors is . (Use a comma to separate vectors as needed.)

Let R2x2 be a linear space with all 2 × 2 real matrices. (1) Find a basis of R2x2 and dim (R²×2); (2) Find the coordinate of the matrix 1 -2 3 under your given basis in (1); (3) If the inner-product on R2x2 is defined by (A,B) = tr(BT A), VA, BER²X2 find an orthonormal basis of R2X2 using the Gram-Schmidt process.