i 4 5 Consider the matrix A = | -1 0 5 -1 3. 1 1 2 Part (a): Determine a basis for the column space, and a basis for the null space, of this matrix. Part (b): State the rank and nullity of this matrix. Part (c): If a linear transformation S: R4 → R³ is defined with this as its standard matrix (so S (x) = Ax for every vector in the domain), is S one-to-one? Is it onto? For each of these, explain why or why not.

Let a € R, and define vectors u₁, U₂, U3 € R³ in terms of a by - 8. a U₁ = 2 U₂ = 3a + 3 2a +3 a² + 3a 7 [4a - 2] 13 = 3a-1 2a² (a) Show that the rank of the matrix [u₁ u₂ u3] does not depend on a. Hint: Find the rank by putting the matrix in row-echelon form. (b) Making reference to a fact from the course, show that u₁, U₂, U3 never span R³, no matter what value a takes. (c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c₂U2. The scalars c₁ and c₂ will depend on a.