Linear 1 T/F

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([T]βα)^−1=[T^−1]βα

False

A vector space cannot have more than one basis.

False

A vector space may have more than one zero vector.

False

AB=I implies that A and B are invertible.

False

A^2=I implies that A=I or A=−I.

False

A^2=O implies that A=O, where O denotes the zero matrix

False

An m×n matrix has m columns and n rows.

False

Every system of linear equations has a solution.

False

Every vector space has a finite basis.

False

Given x1, x2∈V and y1, y2∈W, there exists a linear transformation T:V→W such that T(x1)=y1 and T(x2)=y2.

False

If S generates the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way.

False

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

False

If T is linear, then T carries linearly independent subsets of V onto linearly independent subsets of W.

False

If T is linear, then nullity(T)+rank(T)=dim(W).

False

If T(x+y)=T(x)+T(y), then T is linear.

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

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

False

If m=dim(V) and n=dim(W), then [T]^γ_β is an m×n matrix.

False

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

False

In any vector space, a{x}=a{y} implies that {x}={y}.

False

In any vector space, a{x}=b{x} implies that a=b.

False

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

False

L(V,W)=L(W,V).

False

Let W be the xy-plane in R3; that is, W={(a1,a2,0) :a1,a2∈R}. Then W=R2.

False

M2×3(F) is isomorphic to F^5

False

Subsets of linearly dependent sets are linearly dependent.

False

Suppose that β={x1,x2,...,xn} and β′={x′1,x′2,...,x′n} are ordered bases for a vector space and Q is the change of coordinate matrix that changes β′-coordinates into β-coordinates. Then the jth column of Q is [xj]β′.

False

T is one-to-one if and only if the only vector x such that T(x)=0 is x=0.

False

T=LA for some matrix A.

False

T=LA, where A=[T]βα.

False

The dimension of M_{m×n}(F) is m+n.

False

The dimension of Pn(F) is n.

False

The empty set is a subspace of every vector space.

False

The empty set is linearly dependent.

False

The intersection of any two subsets of V is a subspace of V.

False

The matrices A, B∈Mn×n(F) are called similar if B=Q^tAQ for some Q∈Mn×n(F)

False

The span of ∅ is ∅.

False

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

False

The zero vector space has no basis.

False

[T^2]βα=([T]βα)^2

False

[U(w)]β=[U]βα[w]β for all w∈W

False

[UT]γα=[T]βα[U]γβ

False

A is invertible if and only if LA is invertible.

True

A must be square in order to possess an inverse.

True

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

True

A vector in F^n may be regarded as a matrix in M_{n×1}(F).

True

Ann×n diagonal matrix can never have more than n nonzero entries.

True

Any set containing the zero vector is linearly dependent.

True

Every change of coordinate matrix is invertible.

True

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

True

Every vector space contains a zero vector.

True

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

True

For any scalar a, aT+U is a linear transformation from V to W.

True

If A is invertible, then (A^−1)^−1=A.

True

If A is square and Aij=δijfor all i and j, then A=I.

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.

True

If T is linear, then T preserves sums and scalar products.

True

If T is linear, then T(0V)=0W.

True

If T,U:V→Ware both linear and agree on a basis for V, then T=U.

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 if and only if S spans V.

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 other than the zero vector space, then V contains a subspace W such that W not=V.

True

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

True

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

True

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

True

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

True

L(V,W) is a vector space.

True

LA+B=LA+LB

True

Let T be a linear operator on a finite-dimensional vector space V, let β and β′ be ordered bases for V, and let Q be the change of coordinate matrix that changes β′-coordinates into β-coordinates. Then [T]β=Q[T]β′Q−1.

True

Let T be a linear operator on a finite-dimensional vector space V. Then for any ordered bases β and γ for V, [T] β is similar to [T]γ.

True

Pn(F) is isomorphic to Pm(F) if and only if n=m.

True

Subsets of linearly independent sets are linearly independent.

True

Suppose that V is a finite-dimensional vector space, that S1 is a linearly independent subset of V,andthatS2is a subset of V that generates V. Then S1 cannot contain more vectors thanS2.

True

T is invertible if and only if T is one-to-one and onto.

True

The zero vector is a linear combination of any nonempty set of vectors.

True

Two functions in F(S, F) are equal if and only if they have the same value at each element of S.

True

[IV]α=I.

True

[T(v)]β=[T]βα[v]α for all v∈V.

True

[T+U]γβ=[T]γβ+[U]γβ.

True

[T]^γ_β=[U]^γ_β implies that T=U.

True


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