Discrete Math and Vector Spaces

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What can be said about the set L(A) of all linear combinations of elements of a set A?

It is a subspace. (correct)

It is not a subspace.

It is only closed under addition.

It is not closed under addition and multiplication by scalars.

Why does L(A) include each ai?

Because L(A) contains all linear combinations of elements of A. (correct)

Because L(A) is closed under addition.

Because L(A) is closed under multiplication by scalars.

Because L(A) is a subspace.

What can be said about a subspace W that includes A?

It contains each linear combination L Xiai. (correct)

It does not contain each linear combination L Xiai.

It cannot contain any linear combinations L Xiai.

It only contains some linear combinations L Xiai.

How can a sum of two linear combinations of elements of A be rewritten?

As a finite sum of scalars times elements of A. Signup and view all the answers

What does it mean for a set A to span a vector space V?

L(A) is equal to V. Signup and view all the answers

Is the function space on an infinite set A always finite-dimensional?

No Signup and view all the answers

What is the space of all mappings f: A -> W?

A vector space of functions Signup and view all the answers

What is Hom(V, W)?

A subset of WV consisting of all linear mappings Signup and view all the answers

What is the theorem 3.2 stating?

Hom(V, W) is a vector subspace of WV Signup and view all the answers

Why is Hom(V, W) a subspace?

Because it is closed under addition and scalar multiplication Signup and view all the answers

What is the theorem 3.3 stating?

The composition of linear maps is linear Signup and view all the answers

What is the result of the composition of linear maps?

A linear mapping from V to X Signup and view all the answers

What is the value of L~ at t, given the coefficients xi and functions Ii?

Xl sin t + X2 cos t + X3et Signup and view all the answers

What is the image of ~1 under the linear mapping T?

sin t Signup and view all the answers

What is the set of all real-valued functions on IR?

IRR Signup and view all the answers

What is the theorem that states that every linear mapping from IRn to W is of a certain form?

Theorem 1.2 Signup and view all the answers

What is the coefficient of et in the linear combination L~?

X3 Signup and view all the answers

What is the spanning set for IRn?

{~i} Signup and view all the answers

What is the property of the linear combination map T?

Linearity Signup and view all the answers

What is the result of T(x + y) in terms of T(x) and T(y)?

T(x + y) = T(x) + T(y) Signup and view all the answers

What is the significance of the theorem in this context?

It is a tremendously important theorem Signup and view all the answers

What is the notation La for?

A linear combination mapping from IR to W Signup and view all the answers

What is the skeleton of a linear mapping T?

The n-tuple {T(5i)} Signup and view all the answers

What is the relationship between the map a -> La and the set of all linear maps T from IR to W?

They are bijective Signup and view all the answers

What is the topic of Chapter 3 in the table of contents?

Vector Spaces Signup and view all the answers

Discrete Math and Vector Spaces

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