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Mathematics for Computer Scientists 2 (COMP1045 UNMC) (SPM1 24-25)

Mathematics for Computer Scientists 2 (COMP1045 UNMC) (SPM1 24-25)

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Steinitz' Lemma can be used to show that any spanning set of a finite-dimensional vector space must contain at least as many vectors as:

One less than the dimension of the space

The dimension of the space

Half the dimension of the space

Twice the dimension of the space

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If W is a subspace of V and S ⊆ W, then:

span(S) ∩ W = {0}

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Is the linear mapping T: R^2 -> R^2 defined by T(x, y) = (x + y, x + y) injective?

No, because Ker(T) != {0}.

No, because Im(T) != R^2.

Yes, because Im(T) = R^2.

Yes, because Ker(T) = {0}.

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Consider the vector space of polynomials of degree at most 2. Which set of polynomials is linearly independent?

{x, x^2, 2x - 3x^2}

{1, 1 + x, 1 + x + x^2, x^2}

{1 + x, 2 + 2x, x^2}

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A linear mapping T: V -> W is called an endomorphism if:

V is a subset of W.

dim(V) = dim(W).

W is a subset of V.

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The kernel of the linear mapping T: R^2 -> R defined by T(x, y) = x - y is:

{ (t, -t) in R^2 | t in R }

{ (t, t) in R^2 | t in R }

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For two subspaces W1 and W2 of V, their sum W1 + W2 is a direct sum (denoted W1 ⊕ W2) if and only if

dim(W1) = dim(W2)

W1 ⊆ W2 or W2 ⊆ W1

W1 ∩ W2 = {0}

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Which of the following statements describes the closure under addition axiom for a vector space V?

There exists a vector 0 in V such that for all u in V, u + 0 = u.

For all vectors u and v in V, their sum u + v is also in V.

For every vector u in V, there exists a vector -u in V such that u + (-u) = 0.

For all scalars a and all vectors u in V, the scalar multiple au is in V.

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The axiom for the existence of opposite elements (additive inverses) in a vector space V states that:

For all scalars a and b and all vectors u in V, (a + b)u = au + bu.

There exists a vector 0 in V such that for all u in V, u + 0 = u.

For all scalars a and all vectors u and v in V, a(u + v) = au + av.

For every vector u in V, there exists a vector -u in V such that u + (-u) = 0.

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Gaussian elimination can be used to determine:

The eigenvalues of a matrix

The trace of a matrix

The determinant of a matrix

The rank of a matrix

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