Linear Algebra Quiz

What is a vector space?

• A set of all second-degree polynomials

• A set with vector addition and scalar multiplication satisfying certain axioms

• A set of all integers

• A set of real numbers

Which axiom fails for the set of all integers to be a vector space?

• Axiom 3

• Axiom 6

• Axiom 9

• Axiom 1

What is the definition of a linear combination of vectors?

• A vector that is orthogonal to all other vectors

• A vector that can be expressed as a sum of scalar multiples of other vectors

• A vector with only direction

• A vector with only magnitude

In Example 7.4, what are the values of a, b, and c for the polynomial u=4x^2-2x+5 as a linear combination of given polynomials?

• a=81/70, b=56/35, c=3/35

• a=5, b=6, c=7

• a=8, b=5, c=3

• a=2, b=3, c=5

Which axiom fails for the set of all second-degree polynomials to be a vector space?

• Axiom 3

• Axiom 9

• Axiom 6

• Axiom 1

What is the condition for a set of vectors to be linearly independent?

• The equation Ax = 0 has only the trivial solution

• The vectors must be orthogonal to each other

• The determinant of the matrix formed by the vectors must be zero

• The vectors cannot be written as a linear combination of each other

Which of the following is a property of linear transformations?

• Linear transformations always result in orthogonal vectors

• The sum of two linear transformations is not a linear transformation

• The product of two linear operators is undefined

• The transformation of the sum of two vectors is equal to the sum of the transformations of the vectors

What is the matrix representation of the transformation T(x, y, z) = (2x + z, 4y) with respect to the standard basis?

• [2 1; 0 0]

• [1 2; 0 4]

• [2 0; 0 4]

• [1 0; 0 1]

Which of the following transformations is linear?

• F(x, y, z) = (2x + y, z)

• G(x, y, z) = (4x + z, y - 4z)

• H(x, y, z) = (x + y, 3x - 2z, 2x + 4z)

• T(x, y, z) = (2 + x, y - z + xy)

In the linear transformation L(x, y, z) = (a + b, 2b + 4c), what is the matrix representation with respect to the basis [(9, 2), (2, 1)]?

• [7 2; 0 -5]

• [1 0; 0 1]

• [3 0; 2 4]

• [2 1; 0 0]