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MAS4105 Exam 2 T/F Practice

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Section 1

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[T(v)]subβ = [T]subαupperβ[v]subα for all v ∈ V.

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Mar 1, 2020

Cards (38)

Section 1

(38 cards)

[T(v)]subβ = [T]subαupperβ[v]subα for all v ∈ V.

Front

T

Back

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

Front

F

Back

[U(w)]subβ = [U]subαupperβ[w]subβ for all w ∈ W.

Front

F

Back

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.

Front

F

Back

[UT]subαupperγ = [T]subαupperβ[U]subβupperγ

Front

F

Back

The matrices A, B ∈ Mnxn(F) are called similar if B = QuppertAQ for some Q ∈ Mnxn(F).

Front

F

Back

A² = 0 implies that A = 0, where 0 denotes the zero matrix.

Front

F

Back

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]subβ = Q[T]subβ'Q⁻¹.

Front

T

Back

L(V, W) is a vector space.

Front

T

Back

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

Front

F

Back

T = LsubA, where A = [T]subαupperβ.

Front

F

Back

A² = I implies that A = I or A = -I.

Front

F

Back

T = LsubA for some matrix A.

Front

F

Back

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

Front

F

Back

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

Front

F

Back

If T is linear, then T(0v) = 0w.

Front

T

Back

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

Front

T

Back

[T]subβupperγ = [U]subβupperγ implies that T = U.

Front

T

Back

If m = dim(V) and n = dim(W), then [T]subβupperγ is an m x n matrix.

Front

F

Back

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

Front

F

Back

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

Front

T

Back

If A is square and Asub(ij) = δsub(ij) for all i and j, then A = I.

Front

T

Back

[Isubv]subα = I.

Front

T

Back

[T²]subαupperβ = ([T]subαupperβ)².

Front

F

Back

[T + U]subβupperγ = [T]subβupperγ + [U]subβupperγ.

Front

T

Back

M₂x₃(F) is isomorphic to F⁵.

Front

F

Back

If A is invertible, then (A⁻¹)⁻¹ = A.

Front

T

Back

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

Front

T

Back

A is invertible if and only if LsubA is invertible.

Front

T

Back

AB = I implies that A and B are invertible.

Front

F

Back

Lsub(A+B) = LsubA + LsubB

Front

T

Back

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 [xsubj]subβ'.

Front

F

Back

Every change of coordinate matrix is invertible.

Front

T

Back

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

Front

T

Back

([T]subαupperβ)⁻¹ = [T⁻¹]subαupperβ

Front

F

Back

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

Front

T

Back

A must be square in order to possess an inverse.

Front

T

Back

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

Front

T

Back