26.2. Examples of Vector Spaces

Try these first

1. 1.

   What is ℝⁿ?

   Hint

   Think of how coordinates in two dimensions or three dimensions are represented.

2. 2.

   Name one example of a matrix space.

   Hint

   Consider how matrices are used to organize data.

3. 3.

   What does ℝⁿ represent?

   • Set of all complex numbers
   • Set of all n-tuples of real numbers
   • Set of integers

   Hint

   Think about what 'n' means in dimensions.

4. 4.

   True or False: The set of all polynomials of degree 2 or less with real coefficients forms a vector space.

   • True
   • False

   Hint

   Consider the definition of closure in vector spaces.

5. 5.

   Prove that the set of all 2x2 matrices is a vector space by verifying the axioms.

   Hint

   Start with the basic definitions of matrix addition and scalar multiplication.

6. 6.

   Given a polynomial f(x) = 4 + 2x - x², determine if it can be expressed as a linear combination of the basis {1, x, x²}.

   Hint

   Break down the polynomial into its components and match with the basis.