Linear Algebra Review

Which of the following is false?

• Vector spaces are sets with vector addition and scalar multiplication satisfying some rules

• Vector addition is commutative

• Addition of real numbers distributes to vector addition

• In a vector space, we can multiply any two vectors

Which of the following is false?

• Bases for vector spaces exist

• finite dimensional vector spaces have finite bases

• A useful basis for a vector space may or may not exist

• Bases for vector spaces are unique

Which of the following is false?

• A linear transformation is completely determined by its value on any basis

• A linear transformation between vector spaces takes vector addition to vector addition

• A linear transformation between two vector spaces takes vector addition to scalar multiplication

Which of the following is false?

• All linear transformations between finite dimensional vector spaces are equivalent to multiplication by a matrix

• Composition of linear transformations is given by multiplying the two matrices using matrix multiplication

• The inverse of a linear transformation is given by multiplying by the inverse matrix

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