Problem 10. Let T : P3 → M2×2 be the function defined by T(p) = [p(0) - p(1) ''(2)] . (a) Prove that T is a linear transformation. (b) Find the image of T. (c) Find the kernel of T. Rank-nullity theorem p(0)+p(1) Problem 11. In the exercises of the previous subsection, whenever you had computed one of the kernel/image of the transformation, could you have guessed the dimension of the other? Problem 12. Find all the surjective linear transformations from R³ to R4. Problem 13. Let V, W be finite-dimensional vector spaces. Prove that there exists an injective linear trans- formation L: VW dim(V) ≤ dim(W). Problem 5. (a) Consider a reflection R through the line y = x in R2. Find a basis of R² for which it is easy to describe how R transforms the vectors inside it. Express R as a matrix transformation. (b) Consider a projection P on the line and 固 У = 2x in R2. Draw a picture of the line together with the vectors Describe how P transforms the two vectors. Can you express P as a matrix transformation? Problem 6. Let R denote a counterclockwise rotation by an angle of 0 in R². Express R as a matrix transfor- mation.

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Find the matrix A of the rotation about the z-axis through an angle of pi/2, counterclockwise as viewed from the positive z-axis.

Linear Algebra. Please write your answer clearly. Thanks.

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Write down the 4 x 4 rotation matrix that takes the orthonormal 3D vectors u (Xu, Yu, Zu), V = (xv, Yv, Zv), and w = (xw, Yw, zw), to orthonormal 3D vectors a = (xa, ya, Za), b (xb, Yb, zb), and c (xc, yc, Ze), So Mu - a, Mv= b, and Mw= = C. ****** =

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

198. Apply the 57-degree counterclockwise rotation about the origin to the vectors [1,0] and [0, 1], then use the image vectors (written as columns) to form the coefficient matrix M for the rotation. Test M by calculating the products M and M Where does this rotation send the vector [3, 1]? Does M, when applied to 3 do its job correctly? 6. 6.

Linear Algebra. Please write your answer clearly. Thanks.

Let the rotation matrices be in each of the axes (Shown in the picture) Determine the result of rotating the vector (7, 4, 1) first π /3 around the x-axis, then π/ 4 around the y-axis, and finally π/2 around the z-axis. Note: In the image it is clearer, do not skip any step, to be able to reach the result.