3. Show that the set {(x, y, z) in R^3 | |x| <= 1, lyl <= 1, and |Z| <= 1) is convex.

Show that the distance between the two points z1 and z2 in anArgand diagram is | z1 - z2 | .

5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z) such that %3D |r- rol = 1.

Suppose SCR" is convex. Show that cl(S) is also convex.

Let S ⊂ R3 be the set consisting of 2 points (1, 2, 2) and (1, 2, 3).Find the annihilator of S. Show expliocitly that the annihilator is asubspace. What is the dimension of the annihilator?

Qz/ Find the Paranetric aned standand.equatior of the line Hhat passer through Points (-2,1,0) &(3,5)

10. Let ā = (1,2) and b= (2,1). Find projz(ā). Then show that ā– projz(ā) is orthogonal to b algebraically. Also provide a diagram.

9. S = {(6,2, 1), (–1,3, 2)} is: linearly independent set of R³. b. a linearly dependent set of R³. а. а

X, Y and Z be subsets of the set . Verify the following properties: XU(YUZ) = (XUY)UZ, Xn(Y_Z) = (XnY)nz (Associativi XU(YNZ) = (XUY)N(XUZ), XN(Yuz)=(XNY)u(XNZ) XcY+XUY=Y+X Y=X

Determine whether the following sets are linearly dependent or linearlyindependent. (a) {(; )( )} . -9} in Maxa(R) 4 (b) {(- ).(- 9G -)} in M2x2(R) 4 (c) {x 3 + 2x2, -x2 + 3x + 1, x3 – x2 + 2x – 1} in P3(R) (d) {x 3 – x, 2x2 + 4, -2x3 + 3x2 + 2x + 6} in P3(R) (e) {(1, -1, 2), (1, -2, 1), (1, 1, 4)} in R3 (f) {(1, –1, 2), (2, 0, 1), (–1, 2, –1)} in R3 {(÷ ) ( -) (; 0. 2 ).(- )} in Max2(R) (g) (m) {(-; ') (? ). -2)} in M2x2(R) () (x 4 - х3 + 5x2 - 8х + 6, -х4 + х3 - 5х2 + 5х - 3, x 4 + 3x2 – 3x + 5, 2x4 + 3x3 + 4x2 – x + 1, x3 – x + 2} in P4(R) ) (x4 - х3 + 5x2 - 8х + 6, -х4 + x3 - 5x2 + 5х - 3, х4 + 3x2 - 3х + 5, 2х4 + х3 + 4x2 + 8х} in P4(R)