31 4 Consider R³ with dot product and Euclidean norm, a subspace U, vector x defined below. Compute the orthogonal projection Пlu(x) of x onto U, a distance d(x, U) from x onto U. Include necessary tableau work for linear indepe and/or solving system(s) of linear equations. U = span[u₁ u2= x= 0 5

plz do not use the topics like rank, vector spaces, subspaces, column spaces, etc. or their associated theory to explain part C

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Use the function to find the image of v and the preimage of w. T(V1 V₂ V3) = (V₂ - V₁, V₁ + V₂, 2v₁), V = (4, 3, 0), w = (-11, 3, 14) (a) the image of v (b) the preimage of w (If the vector has an infinite number of solutions, give your answer in terms of the parameter t.) Need Help? Read It Determine whether the function is a linear transformation. T: R² R³, T(x, y) = (2x², xy, 2y²) O linear transformation O not a linear transformation Need Help? Watch It Read It Watch It

Let X=(1,-3)t be the transpose of vector (1,-3). If you rotate X by angle of π/3 what are the components of a new point?

Use the function to find the image of v and the preimage of w. VL, V2) = (v v2 ), v = (9, 9), w = (-3/2, 2, -8) T( V2, V1 + v2, 2v1 2 (a) the image of v (b) the preimage of w (If the vector has an infinite number of solutions, give your answer in terms of the parameter t.)

Let x be the unit vector parallel to the vector from the point (2, 3, −2) to the point (-6,2,2). Find the vector projection of x onto a line with direction vector of (-7,1,2)