MATH 296 PROBLEMS 5

IGOR KRIZ

1. Consider a linear map f : ℝ^{n} → ℝ^{m} fiven by f(x) = Ax, where A is an m × n matrix. Prove that the image of f, i.e. the set {Ax|x ∈ ℝ^{n}}, is equal to the column space Col(A).

2. A problem in Chemistry: The artificial sweetener Aspartame has chemical formula C_{14}H_{18}O_{5}N_{2}.

(a) Is it possible to prepare Aspartame from the following ingredients with no byproducts: Sucrose (C_{12}H_{22}O_{11}), glucose (C_{6}H_{12}O_{6}), water (H_{2}O), nitric acid (HNO_{3}) and nitrogen pentoxide (N_{2}O_{5})? [In chemistry, a formula consists of names of atoms in the molecule; the subscript stands for the number of atoms of an element in the molecule. H is hydrogen, C is carbon, O is oxygen and N is nitrogen. For example, a molecule of water has 2 atoms of hydrogen and one atom of oxygen.]

(b) The molecules in (a) can be represented by elements of ℝ^{4} where the coordinates are quantities of atoms of the different elements C, H, O, N. Find a basis and dimension of the vector space V spanned by the vectors representing the molecules of the ingredients in (a).

3. Which of the following sets are vector subspaces of ℝ^{3}?

(a) U = \{( \begin{matrix}a \\ b \\ 1 \end{matrix} ) |a and b in ℝ\}

(b) U = \{( \begin{matrix}a \\ b \\ c \end{matrix} ) |a + 2b − c = 0, a, b and c in ℝ\}

(c) U = \{( \begin{matrix}a \\ b \\ c \end{matrix} ) |a^{2} + b^{2} = c^{2}, a, b and c in ℝ\}.

4. (a) Is the set

( \begin{matrix}1 \\ −2 \\ 1 \\ 3 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ −1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ −1 \\ 0 \\ 3 \end{matrix} )

linearly independent in ℝ^{4}?

(b) Is the set {x^{2}, e^{x}, sin(x)} linearly independent in the vector space of all functions f : ℝ → ℝ?

5. Prove that the set of all m × n matrices is a vector space where addition is addition of matrices, and scalar multiplication by λ ∈ ℝ is given by multiplying every entry of a matrix by λ.