MATH 240 QUIZ 2 AND CORRECT ANSWERS

MATH 240 QUIZ 2 AND CORRECT
ANSWERS
Dim(A) (dimension of subspace) ✅✅ANSW-the number of vectors in the basis for the subspace



Rank(A) ✅✅ANSW-dimCol(A) -> the number of lin. indep columns in Col(A)



maximum possible rank ✅✅ANSW-the smaller of M and N



rank theorem ✅✅ANSW-If A has n columns: rank(A)+dimNul(A)=n and dimNul(A)+dimCol(A)=n,
and rank is the min{m,n}



equivalent statements ✅✅ANSW-1.A is invertible

2. Col(A)=Rn

3.dimCol(A)=n

4.Rk(A)=n

5.Nul(A)={0}

6.dimNul(A)=0



if V1...Vp are in Rn, then Span {V1...Vp} is the same as the column space of the matrix [V1 Vp]
✅✅ANSW-true



the set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rm
✅✅ANSW-false



the columns of an invertible nxn matrix form a basis for Rn ✅✅ANSW-true



Properties of subspace of Rn ✅✅ANSW-0 vecter is in H, if u and v are in H, u + v is in h, if u is in h
-> lambda*u is in H



Col(A) (column space) ✅✅ANSW-the set of all linear combos of the columns of A

, Nul(A) (null space) ✅✅ANSW-the set of all solutions to Ax=0



Nul(A) is a subspace of Rn ✅✅ANSW-true



Basis for a subspace H ✅✅ANSW-a linearly independent set in H that spans H (the vectors in the
set of all solutions)

row operations do not affect linear dependence relations among the columns of a matrix
✅✅ANSW-true



a subset H of Rn is a subspace if the zero vector is in H ✅✅ANSW-false, there are more conditions
than that



if B is an echelon form of a matrix A, then the pivot columns of B form a basis for ColA ✅✅ANSW-
false, pivot columns of A form the basis



given vectors v1....vp in Rn, the set of all linear combinations of these vectors is a subspace of Rn
✅✅ANSW-true



let H be a subspace of Rn. If x is in H, and y is in Rn, then x+y is in H ✅✅ANSW-false



the column space of a matrix A is the set of solutions of Ax=b ✅✅ANSW-false



If B={V1...Vp} is a basis for a subspace H and if x=C1V1+...+CpVp, then C1...Cp are the coordinates of
x relative to the basis B ✅✅ANSW-true



Each line in Rn is a one-dimensional subspace of Rn ✅✅ANSW-false



the dimension of ColA is the number of pivot columns in A ✅✅ANSW-true



the dimensions of ColA and NulA add up to the number of columns in A ✅✅ANSW-true