Question

1. Let ⃗u = −2[4,0,1]+[−1,3,−2] and ⃗v = 3[4,0,1]+5[−1,3,−2]. Let w⃗ = 3⃗u−⃗v. Express w⃗ as a linear combination of the vectors [4, 0, 1] and [−1, 3, −2].

2. Let ⃗u and ⃗v be two vectors in Rn. Suppose that ||⃗u|| = 3, ||⃗u − ⃗v|| = 5, and that⃗u.⃗v = 1. What is ||⃗v||?.

3. Let ⃗u and ⃗v be two vectors in Rn. Suppose that ||⃗u|| = 5 and that ||⃗v|| = 2. Show that ||⃗u − ⃗v|| ≥ 3. (Hint: one can write ⃗u as (⃗u − ⃗v) − ⃗v).

Answer #1

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

2. Given ⃗v = 〈3,4〉 and w⃗ = 〈−3,−5〉 find
(a) comp⃗vw⃗
(b) proj⃗vw⃗
(c) The angle 0 ≤ θ ≤ π (in radians) between ⃗v and w⃗.
1. Let d--> =2i−4j+k. Write⃗a=3i+2j−6k as the sum
of two vectors⃗v,w⃗, where⃗v is perpendicular
to d--> and w⃗ is parallel to
d-->.

Let u = ⟨1,3⟩ and v = ⟨4,1⟩.
(a) Find an exact expression and a numerical approximation for
the angle between u and v. (b) Find both the projection of u onto v
and the vector component of u orthogonal to v.
(c) Sketch u, v, and the two vectors you found in part
(b).

Let u = (−3, 2, 1, 0), v = (4, 7, −3, 2), w = (5, −2, 8, 1).
Find the vector x that satisfies 2u − ||v||v = 3(w − 2x).

Let U, V be a pair of subspaces of Rn and U +V the
summationspace. Suppose that U ∩ V = {0}. Prove that from every
vector U + V can be written as the sum of a vector from U and a
vector from V.

5.
Perform the following operations on the vectors u=(-4, 0, 2),
v=(-1, -5, -4), and w=(0, 3, 4)
u*w=
(u*v)u=
((w*w)u)*u=
u*v+v*w=

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4,
∥u + v∥ = 5. Find the inner product 〈u, v〉.
(b) Suppose {a1, · · · ak} are orthonormal vectors in R m.
Show that {a1, · · · ak} is a linearly independent set

Let u, v, and w be vectors in Rn. Determine which of the
following statements are always true. (i) If ||u|| = 4, ||v|| = 5,
and ?||u + v|| = 8, then u?·?v = 4. (ii) If ||u|| = 2 and ||v|| =
3, ?then |u?·?v| ? 5. (iii) The expression (v?·?w)u is both
meaningful and defined. (A) (ii) and (iii) only (B) (ii) only (C)
none of them (D) all of them (E) (i) only (F) (i) and...

Let T be a linear transformation that is one-to-one, and u, v be
two vectors that are linearly independent. Is it true that the
image vectors T(u), T(v) are linearly independent? Explain why or
why not.

1)
a) Let z=x4 +x2y, x=s+2t−u, y=stu2:
Find:
( I ) ∂z ∂s
( ii ) ∂z ∂t
( iii ) ∂z ∂u
when s = 4, t = 2 and u = 1
1) b> Let ⃗v = 〈3, 4〉 and w⃗ = 〈5, −12〉. Find a
vector (there’s more than one!) that bisects the angle between ⃗v
and w⃗.

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