anonymous 4 years ago determine bases for the following subspaces of R^3 a) the plane 3x-2y+5z

1. anonymous

Thank you, luck :^)

2. anonymous

sir its my pleasure that you are helping me

3. anonymous

That plane 3x - 2y + 5x = 0 is a two dimensional subapace of $\mathbb{R}^{3} .$ If we solve for z in terms of x and y, we get z = (-3x + 2y)/5. Letting x = 1, y = 0 makes z = -3/5. Letting x = 0, y = 1 makes z = 2/5, so two linearly independent vectors in the subspace are [1,0,-3/5] and [0,1, 2/5]. Those two vectors form a basis for the subspace $B = \left\{ [x, y, z]|3x - 2y + 5z = 0 \right\}.$

4. anonymous

Consider the operation on p2 that takes ax^2+bx+c to cx^2+bx+a . Does it correspond to a linear transformation from R^3 to R^3 ? If so, what is its matrix? (b) Consider the operation on p3 that takesax^3+bx^2+cx+d to cx^3-bx^2-ax+d . Does it correspond to a linear transformation from R^3 to R^3 ? If so, what is its matrix?

5. anonymous

can you solve above question

6. anonymous

Someone help on my problems.

7. anonymous

Let us consider n, a unit normal vector to the plane at the origin, and $u_1 ,$ the unit vector in the direction of [1, 0 ,-3/5]. $n = [3, -2, 5]/\sqrt{3^{2} + (-2)^{2} + 5^{2},}$ $u_1 = [5/\sqrt{34}, 0 -3/\sqrt{34}].$ If we take the cross product of unit normal vector N with $u_1 ,$we get another unit vector $u_2$in the subspace S, so those two vectors form another basis if S; in fact, it is an orthonormal basis of S. To determine what $u_2$is, evaluate the determinant of matrix [[ i, j, k ], [3/sqrt(38), -2/sqrt(38), 5/sqrt(38)], [5/sqrt(34), 0, -3/sqrt(34)]] = [6/sqrt(1292), 34/sqrt(1292), 10/sqrt(1292)], simplifying to [3/ sqrt(323), 17/sqrt(323), 5/sqrt(323) ]. Thus, an orthonormal basis for subspace S is$\left\{ u_1, u_2 \right\},$where $u_1 = [5/\sqrt{34}, 0, -3/\sqrt{34}],$ $u_2 = [3/\sqrt{323}, 17/\sqrt{323}, 5/\sqrt{323}].$

8. anonymous

solve this please (b) Consider the operation on p3 that takes ax^3+bx^2+cx+d to cx^3-bx^2-ax+d . Does it correspond to a linear transformation from R^3 to R^3 ? If so, what is its matrix?

9. anonymous

luck you should be asking one question at a time as a new one so he can get more medals for such awesome answers ;D

10. anonymous

Thank you, luck :^)

11. anonymous

sir can you solve the question i posted recently

12. anonymous

Let me see can I find it...