Vector Spaces and Scalar Multiplication Quiz

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What is the extra structure of multiplication by a scalar called on an abelian group?

Scaling (correct)

Real numbers

Addition

Binary operation

What is the binary operation defined on R3?

Subtraction

Multiplication

Addition (correct)

Division

How is the operation (α, v) denoted?

v*α

α.v (correct)

α*v

v/α

What is the result of α.(β.v) according to the given example?

(αβ).v Signup and view all the answers

What is the abelian group (R3, ∗) together with the rule of scaling called?

Vector space Signup and view all the answers

What is the defining property of the scaling operation on R3?

The scaling operation satisfies the properties: 1. α.v = v, 2. α.(β.v) = (αβ).v, 3. α.(v ∗ w) = α.v ∗ α.w, 4. (α + β).v = α.v ∗ β.v Signup and view all the answers

How is the operation (α, v) denoted?

The operation (α, v) is denoted by α.v Signup and view all the answers

What are the properties of the scaling operation on R3?

The properties of the scaling operation on R3 are: 1. α.v = v, 2. α.(β.v) = (αβ).v, 3. α.(v ∗ w) = α.v ∗ α.w, 4. (α + β).v = α.v ∗ β.v Signup and view all the answers

What is the name of the abelian group (R3, ∗) together with the rule of scaling?

The abelian group (R3, ∗) together with the rule of scaling is called a vector space. Signup and view all the answers

What are the defining properties of the scaling operation on R3?

The defining properties of the scaling operation on R3 are: 1. α.v = v, 2. α.(β.v) = (αβ).v, 3. α.(v ∗ w) = α.v ∗ α.w, 4. (α + β).v = α.v ∗ β.v Signup and view all the answers