Question

3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F with dim(V) = n and dim(W) = m, and

let φ : V → W be a linear transformation. Fill in the six blanks to give bounds on the sizes of the

dimension of ker(φ) and the dimension of im(φ).

3. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F with dim(V ) = n and dim(W) = m, and

let φ : V → W be a linear transformation. Fill in the six blanks to give bounds on the sizes of the

dimension of ker(φ) and the dimension of im(φ).

max(____,_____) ≤ dim(ker(φ)) ≤_____ _____≤ dim(im(φ)) ≤ min(_____,_____)

Answer #1

. and . Then,

Of course, . and also . Because the subspce is always subset of . Also, can send at max whole space to . So .

, because can be the map. Also, . Image can not exceed the space .

1. Let V and W be ﬁnite-dimensional vector spaces over ﬁeld F
with dim(V) = n and dim(W) = m, and
let φ : V → W be a linear transformation.
A) If m = n and ker(φ) = (0), what is im(φ)?
B) If ker(φ) = V, what is im(φ)?
C) If φ is surjective, what is im(φ)?
D) If φ is surjective, what is dim(ker(φ))?
E) If m = n and φ is surjective, what is ker(φ)?
F)...

2. For ﬁnite-dimensional vector spaces V and W over ﬁeld F, let
φ : V → W be a linear transformation.
A) If φ : R^83 → M(7 × 9, R) is surjective, what is
dim(ker(φ))?
B) If φ : P24(R) → P56(R) is injective, what is dim(im(φ))?
C) If φ : M(40 × 3, R) → R^71 has dim(im(φ)) = 67, what is
dim(ker(φ))?
D) If φ : R^92 → P105(R) has dim(ker(φ)) = 8, what is
dim(im(φ))?

Let V and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

Let L : V → W be a linear transformation between two vector
spaces. Show that dim(ker(L)) + dim(Im(L)) = dim(V)

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

Construct a linear transformation T : V → W, where V and W are
vector spaces over F such that the dimension of the kernel space of
T is 666. Is such a transformation unique?
Give reasons for your answer.

Suppose V and W are two vector spaces. We can make the set V × W
= {(α, β)|α ∈ V,β ∈ W} into a vector space as follows:
(α1,β1)+(α2,β2)=(α1 + α2,β1 + β2)
c(α1,β1)=(cα1, cβ1)
You can assume the axioms of a vector space hold for V × W
(A) If V and W are finite dimensional, what is the dimension of
V × W? Prove your answer.
Now suppose W1 and W2 are two subspaces of V ....

Let V be an n-dimensional vector space. Let W and W2 be unequal
subspaces of V, each of dimension n - 1. Prove that V =W1 + W2 and
dim(Win W2) = n - 2.

) Let L : V → W be a linear transformation between two finite
dimensional vector spaces. Assume that dim(V) = dim(W). Prove that
the following statements are equivalent. a) L is one-to-one. b) L
is onto.
please help asap. my final is tomorrow morning. Thanks!!!!

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 23 minutes ago

asked 31 minutes ago

asked 38 minutes ago

asked 40 minutes ago

asked 53 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago