Linear Algebra

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
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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
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
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