Question

Linear Algebra question: Prove that if A:X→Y and V is a subspace of X then dim...

Linear Algebra question: Prove that if A:X→Y and V is a subspace of X then dim AV ≤ dim V. Deduce from here that rank(AB) ≤ rank B

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Linear Algebra: Show that the set of all 2 x 2 diagonal matrices is a subspace...
Linear Algebra: Show that the set of all 2 x 2 diagonal matrices is a subspace of M 2x2. I know that a diagonal matrix is a square of n x n matrix whose nondiagonal entries are zero, such as the n x n identity matrix. But could you explain every step of how to prove that this diagonal matrix is a subspace of M 2x2. Thanks.
Let U be a vector space and V a subspace of U. (a) Assume dim(U) <...
Let U be a vector space and V a subspace of U. (a) Assume dim(U) < ∞. Show that if dim(V ) = dim(U) then V = U. (b) Assume dim(U) = ∞ and dim(V ) = ∞. Give an example to show that it may happen that V 6= U.
Linear algebra question Show that T = {y ∈ C∞(R) : y00−49y = 0} is a...
Linear algebra question Show that T = {y ∈ C∞(R) : y00−49y = 0} is a vector subspace of C∞(R).
Suppose V is a vector space over F, dim V = n, let T be a...
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
Let V be the subspace of all vectors in R 5 , such that x1 −...
Let V be the subspace of all vectors in R 5 , such that x1 − x4 = x2 − 5x5 = 3x3 + x4 (a) Find a matrix A with that space as its Null space; What is the rank of A? b) Find a basis B1 of V ; What is the dimension of V ? (c) Find a matrix D with V as its column space. What is the rank of D? To find the rank of...
1) Let T : V —> W be a linear transformation with dim(V) = m and...
1) Let T : V —> W be a linear transformation with dim(V) = m and dim(W) = n. For which of the following conditions is T one-to-one? (A) m>n (B) range(T)=Wandm=n (C) nullity(T) = m (D) rank(T)=m-1andn>m 2) For which of the linear transformations is nullity (T) = 0 ? Why? (A)T:R3 —>R8 (B)T:P3 —>P3 (C)T:M23 —>M33 (D)T:R5 —>R2 withrank(T)=2 withrank(T)=3 withrank(T)=6 withrank(T)=1
4. Prove the Following: a. Prove that if V is a vector space with subspace W...
4. Prove the Following: a. Prove that if V is a vector space with subspace W ⊂ V, and if U ⊂ W is a subspace of the vector space W, then U is also a subspace of V b. Given span of a finite collection of vectors {v1, . . . , vn} ⊂ V as follows: Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in the scalar field}...
Hi. I have two questions about the linear algebra. 1. Prove that a linear transform always...
Hi. I have two questions about the linear algebra. 1. Prove that a linear transform always maps 0 to 0. 2. Suppose that S = {x, y, z} is a linearly dependent set. Prove that every vector v in the span of the set S can be expressed as a linear combination in more than one way. Will thumb up for both answers. Thank you so much!
1.a Implication: If an endomorphism f : V → V is not an isomorphism, then it...
1.a Implication: If an endomorphism f : V → V is not an isomorphism, then it must have a non-trivial kernel. Use this implication, together with the definition of determinant given in class, to show that if for an endomorphism f : V → V we have det(f) 6 /= 0, then in fact f is an isomorphism. 1.b. The Rank-Nullity Theorem applies to a linear map f : V → W (where V is finite dimensional) and claims that:...
Consider the subspace S = {[x, y, 2x + 3y] | x, y ∈ R} of...
Consider the subspace S = {[x, y, 2x + 3y] | x, y ∈ R} of R 3 . (a) Find a basis of S and dim (S). (b) Extend the basis of S in (a) to a basis of R 3 .