Question

Let u=(-1, 2, 4)^{T} and v=(4, a, 1)^{T} . For
what value of "a" are these vectors linearly
independent?

**Insert** the value of "a".

Answer #1

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.

Let (u,v,w,t) be a linearly independent list of vectors in R4.
Determine if (u, v-u, w+5v, t) is a linearly independent list.
Explain your reasoning and Show work.

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in 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

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 −...

Let u, vand w be linearly dependent vectors in a vector space V.
Prove that for any vector z in V whatsoever, the vectors u, v, w
and z are linearly dependent.

Let S={u,v,w}S={u,v,w} be a linearly independent set in a vector
space V. Prove that the set S′={3u−w,v+w,−2w}S′={3u−w,v+w,−2w} is
also a linearly independent set in V.

Let U and W be subspaces of a nite dimensional vector space V
such that U ∩ W = {~0}. Dene their sum U + W := {u + w | u ∈ U, w ∈
W}.
(1) Prove that U + W is a subspace of V .
(2) Let U = {u1, . . . , ur} and W = {w1, . . . , ws} be bases
of U and W respectively. Prove that U ∪ W...

Let the linear transformation T: V--->W be such that T (u) =
u2 If a, b are Real. Find T (au + bv) ,
if u = (x, y) v = (z, w) and uv = (xz-yw, xw + yz)
Let the linear transformation T: V---> W be such that T (u)
= T (x, y) = (xy, 0) where u = (x, y), with 2, -3. Then, if u = (
1.0) and v = (0.1). Find the value...

1. Let T be a linear transformation from vector spaces
V to W.
a. Suppose that U is a subspace of V,
and let T(U) be the set of all vectors w in W
such that T(v) = w for some v in V. Show that
T(U) is a subspace of W.
b. Suppose that dimension of U is n. Show that
the dimension of T(U) is less than or equal to
n.

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