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Assignment 4

1. Let v1 = [1, 0, 1, 0], v2 = [0, 1, 0, 2], v3 = [2, 0, 0, 1], and v4 = [0, −2, −3, 0]. Find k1, k2, k3, and k4 such that

k1v1 + k2v2 + k3v3 + k4v4 = [−4, 1, −5, 5].

2. Find u · v if ∥u + v∥ = 1 and ∥u − v∥ = 5.

3. Let u, v ∈ Rn. Show that

4. Let u1, u2, . . . , un ∈ R n . Show that if u1, u2, . . . , un are pairwise orthogonal, i.e., ui · uj = 0 for any i ≠ j, then

5. Show that

6. Find the standard matrix for each of the following linear transformations.

7. Let T : R 3 → R 3 be the linear transformation that counterclockwise rotates each vector about the positive y-axis through an angle θ, where 0 ≤ θ ≤ π/2.

(a) Find the standard matrix for T.

(b) Find the rotation of the vector x = [−5, 1, 2]T through an angle of θ = π/3.