Chapter 4 Vector Spaces

Define isomorphism

A one-to-one linear transformation from some vector space V to another vector space W.

Define what a basis is.

A set of vectors for a subspace H is a basis for H if 1) The is linearly independent 2) The set spans H

Define vector space. (List the ten axioms that must hold)

A vector space is a set of objects, called vectors, that have properties described by a the following ten axioms. (pg. 192)

What is the dimension of the zero vector space?

By definition, it is zero.

Define finite dimensional, dimension, and infinite-dimensional

Finite dimensional means it only takes a finite number of vectors to span it. Inifinite dimensional menas it takes an infinite number of vectors to span it. (pg. 228)

Define subspace (Three properties that subspaces must have)

If H is a subset of a vector space V, then H is a subset if 1) The zero vector of V is in H. 2) H is closed under vector addition. 3) H is closed under scalar multiplication (pg. 195)

Theorem 13 State and Prove. If two matrices are row equivalent, then their row spaces...

If two matrices are row equivalent, then their row spaces are the same. If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B. (pg. 233)

What is the coordinate vector of x (relative to basis B)? Where does it live?

It lives in R^n (pg. 218)

What does it mean for a set of vectors to be linearly independent?

It means that the homogenous vector equation has only the trivial solution

What does ti mean for a set of vectors to be linearly dependent?

It means that the homogenous vector equation has trivial solutions

When is a set containing a single vector linearly independent? When is it linearly dependent?

It will be linearly independent if the vector is not the zero vector. If it is the zero vector, the set will be linearly dependent

Prove that the zero vector is unique (exercise 25 pg. 199)

Suppose that a vector, w, in a vector space V has the property that u+w=w+u=u where u is any vector in V. In particular, 0+w=0 where 0 is the zero vector in V. But 0+w=w by axiom four. Hence w=0+w=0 and we see that the zero vector is unique

What is the coordinate mapping (determined by a basis B)?

The coordinate mapping is a transformation T: V→R^n s.t. V(x)=[x]b (pg. 218)

What is meant by kernal and range?

The kernal is the null space of a linear transformation T. The set of all u in T s.t. T(u)=0. The range of T is the set of all T(x) for some x in V.

Theorem 2. Prove it

The null space of an mxn matrix A is a subspace of R^n. Equivalently, the set of all solutions to a system Ax=0 of m homogeneous linear equations in n unknowns is a subspace of R^n (pg.

Define the rank of a matrix A

The rank of a matrix A is the dim(ColA). "The dimension of the column space of A." Equivalently, the dimension of the row space of A is the rank of a matrix A. We can find the rank of a matrix A by finding the number of pivot columns.

What is the zero subspace?

The set consisting of only the zero vector in a vector space V (pg. 195)

Define column space

The set of all linear combinations of a matrix A, written ColA (pg. 203)

Define row space

The set of all linear combinations of the rows of some mxn matrix A. Denoted as RowA. The rowspace is a subspace of R^n (pg. 233)

Define standard basis

The set of vectors containing the columns of the nxn identity matrix form the standard basis for R^n

Theorem 3. Prove it

This follows immediately from theorem 1 and definition of ColA (pg. 203)

T/F. The dim(RowA)=dim(ColA^T)

This is true because the rowspace of some matrix A is the same thing as the columns space of the transpose of that same matrix

True or false: every subspace is a vector space. Why?

This is true.

Theorem 9. State and Prove

This tells us that if we have a Vector space V with a basis with dimension n, then any set of vectors in V is linearly dependendent if it has more than n vectors. To prove this, we use the fact that the coordinate mapping is linear (pg. 227)

Theorem 14 "The Rank Theorem"

This theorem tells us that the dimensions of the columns space and the row space of a matrix A are the same (both are the rank of A)

Prove that c0=0 for any scalar c and where 0 is the zero vector in some vector space V (see picture)

a) axiom 4 b) axiom 7 c) axiom 3 d) axiom 5 e) axiom 4 (exercise 28 pg. 199)

Redo problem 27 in 4.1. Fill in the blanks

a) axiom 8 b) axiom 3 c) axiom 5 d) axiom 4 (pg. 199)

Theorem 4. An indexed set of vectors is linearly dependent iff ...

at least on the vectors is a linear combination of the others

Prove the following facts using the ten axioms of a vector space

pg. 193

Theorem 1: The span of any vectors in a subspace, V, is a subspace. Prove it.

pg. 196

Define null space

pg. 201

Definition of linear transformation

pg. 206

Define the coordinates of x in V relative to the basis, B

pg. 218

Theorem 7: The unique representation theorem. State and prove.

pg. 218

Theorem 8. State and Prove

pg. 221

What is the change of coordinates matrix?

pg. 221

Redo problem 23 in 4.4

pg. 225

Redo problem 24 in 4.4

pg. 225

Theorem 10. State and Prove

pg. 227

Theorem 11. State and Prove

pg. 229

How does one find the dimensions of NulA and ColA?

pg. 230

Theorem 12. State and Prove

pg. 230

Redo problem 32

pg. 232