consists of a set on which addition and scalar multiplication are defined

vector space

If x, y, and z are vectors in a vector space V such that x + z = y + z, then x = y.

Cancellation Law of Addition

A subset W of a vector space V over a field F is called a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V

subspace

1.closed under addition

2. closed under scalar multiplication

3. there is a zero vector

4. each vector has an additive inverse

2. closed under scalar multiplication

3. there is a zero vector

4. each vector has an additive inverse

A subset of a vector space is a subspace iff:

True

Every vector space contains a zero vector (T/F)

False

A vector space my have more than one zero vector (T/F)

False

In any vector space, ax = bx implies a = b.

False

In any vector space, ax = ay implies that x = y.

True

A vector in F^n may be regarded as a matrix in M(nx1)(F).

False

An mxn matrix has m columns and n rows.

False

In P(F), only polynomials of the same degree may be added

False

If f and g are polynomials of degree n, then f + g is a polynomial of degree n

True

If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n.

True

A nonzero scalar of F may be considered to be a polynomial in P(F) having degree zero

True

Two function in F(S,F) are equal iff they have the same value at each element of S

False

If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V

False

The empty set is a subspace of every vector space

True

If V is a vector space other than the zero vector space, then V contains a subspace W such that W is not equal to V

False

The intersection of any two subset of V is a subspace of V

True

An nxn diagonal matrix can never have more than n nonzero entries

False

The trace of a square matrix is the product of its diagonal entries

False

Let W be the xy-plane in R^3; that is, W = {(a1, a2, 0) : a1, a2 in R}: Then W = R^3

the set consisting of all linear combinations of the vectors in S…

span

True; Thm. 1.5

The span of any subset S of a vector space V is a subspace of V (T/F)

span(S) = V. The vectors of S span V

A subset S of a vector space V spans V iff:

True

The zero vector is a linear combination of any nonempty set of vectors (T/F)

False

The span the the empty set is the empty set

True

If S is a subset of a vector space V, then span(S) equals the intersection of all subspaces of V that contain S

False

In solving a system of linear equations, it is permissible to multiply an equation by any constant

True

In solving a system of linear equations, it is permissible to add any multiple of one equation to another

False

Every system of linear equations has a solution

the sum of the diagonal entries of an nxn matrix

trace

if the matrix = 0 when the rows and columns are not the same

diagonal matrix

if there exists a finite number of distinct vectors in S and scalars not all zero such that their linear combination = 0.

linearly dependent

the empty set, a set consisting of a single nonzero vector, iff the only representations of 0 as lin. combs. of its vectors are trivial (all equal each other)

linearly independent

False

If S is a linearly dependent set, then each vector in s is a linear combination of other vectors in S.

True

Any set containing the zero vector is linearly dependent

False

The empty set is linearly dependent

False

Subsets of linearly dependent sets are linearly dependent

True

Subsets of linearly independent sets are linearly independent

True

If a1x1 + a2x2 + … + anxn = 0 and x1, x2,…, xn are linearly independent, then all the scalars a are zero

False

The zero vector has no basis

True

Every vector space that is generated by a finite set has a basis

False

Every vector space has a finite basis

False

A vector space cannot have more than one basis

True

If a vector space has a finite basis, then the number of vectors in every basis is the same

False

The dimension of Pn(F) is n

False

The dimension of a matrix with m rows and n columns is m + n

True

suppose that V is a finite-dimensional vector space, that S1 is linearly independent subset of V, and that S2 is a subset of V that generates V. Then S1 cannot contain more vectors than S2

False

if s generates the vector space V, then ever vector in V can be written as a linear combination of vectors in S in only one way

True

Every subspace of a finite-dimensional space is finite-dimensional

True

If v is a vector space having dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimension n

True

If V is a vector space having dimension n, and if S is a subset of V with n vectors, then S is linearly independent iff S spans V

linearly independent subset of a vector space that spans the vector space

basis

True; thm 1.9

If a vector space has a finite span, then some subset of that span is a basis for V

a vector space that has a basis consisting of a finite number of vectors

finite-dimensional

unique number of vectors in each basis

dimension