Understanding Vector Spaces and Dimensions

What is a basis of a vector space?

• A set of vectors that spans the space but is linearly dependent

• A set of vectors that are linearly independent but do not span the space

• A set of vectors that are linearly dependent

• A set of vectors that spans the space and is linearly independent

If a vector space has a basis with n vectors, what can be said about any set with more than n vectors?

• It must be linearly dependent

• It forms another basis

• It cannot span the space

• It must be linearly independent

What is the dimension of the vector space R2?

• 1

• 2

• 3

• 4

How many vectors are in the standard basis for the vector space R3?

• 3

• 2

• 1

• 4

What determines the dimension of the vector space of M by N matrices?

• The number of rows

• The number of columns

• The product of the number of rows and columns

• The sum of the number of rows and columns

In the context of matrices, what is a standard basis vector?

• A matrix with all entries zero

• A matrix with one entry as one and the rest as zero

• A matrix with all entries one

• A matrix with diagonal entries as one

What is the first step in finding the dimension of the solution space of a homogeneous linear system?

• Identifying the free variables

• Performing Gauss-Jordan elimination

• Assigning arbitrary parameters

• Counting the number of equations

In Gauss-Jordan elimination, what are leading variables?

• Variables that are free

• Variables that correspond to leading ones in the matrix

• Variables that are arbitrary

• Variables that are set to zero

How do you determine if vectors form a basis for the solution space?

• They must be scalar multiples of each other

• They must have the same number of components

• They must be linearly dependent

• They must be linearly independent

What is the dimension of the solution space if there are two free variables?

• 4

• 3

• 2

• 1