Given the following basis B and vector space V use the Gram-Schmidt procedure to find the orthonormal basis. = {(1, 2, 1), (1, 1, 3) (2, 1, 1)}, V = R³ B = Check if the following mapping is linear transformation or not. Justify your answer T: R² → R³ T(x, y) = (5x, x - 2y, y - x)

Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.

Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.

Let T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).

Find the projection of the vector v=[102]T onto the subspace S=span{[011],[011]}.

Let B= (b1,b2,b3}, D=(d,,d2} be bases for vector spaces V and W, respectively. Let T:V---> W be a linear transformation with the following properties. T(b,) = - 6d1 +7dz, T(b2) = d1-8d2, T(b3) = - d1 Then the matrix from B to D: Po+B -6 7 - 1 -6 1 - 1 1 -8 0 -1 0 a. b. 7 -8 0 -6 1 -6 7 7 -8 0 0 0 0 1 -8 -1 0 C. d.