Math 461

Definitions for Chapter 4

1. A subspace of a vector space V is any set H in V with the properties: _____ a. The zero vector of V is in H.
   b. For each u and v in H, the sum u + v is in H.
   c. For each u in H and each scalar c, the vector cu is in H.
2. The null space of a matrix A is _____ the set of all solutions to the equation Ax = 0.
3. The column space of a matrix A is _____ the set of all linear combinations of the columns of A.
4. A transformation T  from a vector space V into a vector space W is linear if: _____ i. T(u + v) = T(u) + T(v)    for all u and v in V, and
   ii. T(cu) = cT(u)                 for all u in V and all scalars c.
5. An indexed set of vectors {v₁, . . . , v_p} in a vector space V is linearly independent if _____ the vector equation c₁v₁ + · · ·  + c_pv_p = 0 has only the trivial solution.
6. An indexed set of vectors {v₁, . . . , v_p} in a vector space V is linearly dependent if _____ there exist weights c₁, . . . , c_p, not all zero, such that c₁v₁ + · · ·  + c_pv_p = 0.
7. A linear dependence relation among vectors v₁, . . . , v_p in a vector space V is _____ an equation of the form c₁v₁ + · · ·  + c_pv_p = 0, where not all of the c_i are zero.
   (or, "where at least one weight c_i is nonzero.")
8. A basis for a subspace H of Rⁿ is _____ a linearly independent set in H that spans H.
9. The dimension of a vectors space H is _____ the number of vectors in any basis for H.
10. The rank of a matrix A is _____ the dimension of the column space of A. 
    (Note: The rank of A equals the number of pivot columns of A, but this is not the definition of rank A.