Question

Let B be a (finite) basis for a vector space V. Suppose that v is a...

Let B be a (finite) basis for a vector space V. Suppose that v is a vector in V but not in B. Prove that, if we enlarge B by adding v to it, we get a set that cannot possibly be a basis for V. (We have not yet formally defined dimension, so don't use that idea in your proof.)

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be...
Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be a set of inner products on V that is (when viewed as a subset of B(V)) linearly independent. Prove that S must be finite e) Exhibit an infinite linearly independent set of inner products on R(x), the vector space of all polynomials with real coefficients.
Let V be a finite-dimensional vector space and let T be a linear map in L(V,...
Let V be a finite-dimensional vector space and let T be a linear map in L(V, V ). Suppose that dim(range(T 2 )) = dim(range(T)). Prove that the range and null space of T have only the zero vector in common
Let V be an n-dimensional vector space and W a vector space that is isomorphic to...
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...
1. Suppose that ? is a finite dimensional vector space over R. Show that if ???(?...
1. Suppose that ? is a finite dimensional vector space over R. Show that if ???(? ) is odd, then every ? ∈ L(? ) has an eigenvalue. (Hint: use induction). (please provide a detailed proof) 2. Suppose that ? is a finite dimensional vector space over R and ? ∈ L(? ) has no eigenvalues. Prove that every ? -invariant subspace of ? has even dimension.
Suppose we have a vector space V of dimension n. Let R be a linearly independent...
Suppose we have a vector space V of dimension n. Let R be a linearly independent set with order n−2. Let S be a spanning set with order n+ 2. Outline a strategy to extend R to a basis for V. Outline a strategy to pare down S to a basis for V .
5. Let V be a finite-dimension vector space and T : V → V be linear....
5. Let V be a finite-dimension vector space and T : V → V be linear. Show that V = im(T) + ker(T) if and only if im(T) ∩ ker(T) = {0}.
Let V be a vector space of dimension n > 0, show that (a) Any set...
Let V be a vector space of dimension n > 0, show that (a) Any set of n linearly independent vectors in V forms a basis. (b) Any set of n vectors that span V forms a basis.
(3) Let V be a finite dimensional vector space, and let T: V® V be a...
(3) Let V be a finite dimensional vector space, and let T: V® V be a linear transformation such that rk(T) = rk(T2). a) Show that ker(T) = ker(T2). b) Show that 0 is the only vector that lies in both the null space of T, and the range space of T
Let V be a vector space with ordered basis B = (b1, . . . ,...
Let V be a vector space with ordered basis B = (b1, . . . , bn). Does the basis having n elements imply that V is the coordinate space R^n?
Let B={u1,...un} be an orthonormal basis for inner product space V and v b any vector...
Let B={u1,...un} be an orthonormal basis for inner product space V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT