LINEAR ALGEBRA EXAM #3
QUESTIONS WITH CORRECT ANSWERS
Nul A - Answer--set of all solutions of the homogeneous equation Ax=0.
-Nul A = {x:x is in R^n and Ax=0}
-subspace of R^n
-orthogonal complement of Row A
-Nul A^T is the orthogonal complement of Col A
Prove Nul A is a subspace of R^n - Answer-(0) Nul A is a subset of R^n because A has
n columns
(1) 0 of R^n is in Nul A, when x = 0
(2) let u and v be vector in Nul A, Au=0 and Av=0, A(u+v) = Au + Av = 0+0=0
(3) let c be any scalar, A(cu)=c(Au)=c(0)=0
Col A - Answer--Set of all linear combinations of the columns of A. If A=[a1,...,an] then
Col A = Span{a1,...,an}
-Col A = {b: b=Ax for some x in R^n}
-Col A is a subspace of R^m
-Nul A^T is the orthogonal complement of Col A
Col A is all of R^m if and only if? - Answer-the equation Ax=b has a solution for each b
in R^m
Basis - Answer-Let H be a subspace of a vector space V, an indexed set of vetors B=
{b1,...,bp} in V is a basis for H if
1) B is a linearly independent set
2) the subspace spanned by B coinsides with H, i.e., H= Span{b1,..,bp}
-If a vector space V has a basis of n vectors, then every basis of V must consist of
exactly n vectors
, each X in V can be represented as - Answer-c1b1+...+cnbn where B={b1,...,bn} is a
basis for V
Coordinates of x relative to the basis B (b-coordiantes of x) - Answer-B={b1,...,bn} is a
basis for V and x is in V. The B-coordinaets of x are the weights c1,...,cn such that x=
c1b1+...+cnbn
[x]B = [c1...cn] (Vector)
change of coordinates matrix - Answer--Pb=[b1,...,bn]
-Pb[x]B=x
-Pb^-1x=[x]B
DimV - Answer--DimV=number of vectors in a basis for V if V is finite dimensional (V is
spanned by a finite set)
-V is infinitely dimensional if V is not spanned by a finite set.
-dimension of 0 vector space is 0.
dim Nul A - Answer-number of free variables
dim Col A - Answer-number of pivot columns = rank A
row A - Answer--Set of all linear combinations of the row vectors
-subspace of R^n
-Row A = Col A^T
-If two matrices A and B are row equivalent, then their row spaces are the same. If B is
in echelon form, the nonzero entries of B form a basis for the row space of A as well as
that of B.
-Nul A is orthogonal complement of Row A
basis of Row A - Answer-nonzero rows of echelon form
rank A - Answer-dim Col A = dim Row A
Kernel - Answer--Kernel of a linear transformation T from a vector space V into a vector
space W is the set of all u in V such that T(u)= 0 (the zero vector in W)
-Nul A
QUESTIONS WITH CORRECT ANSWERS
Nul A - Answer--set of all solutions of the homogeneous equation Ax=0.
-Nul A = {x:x is in R^n and Ax=0}
-subspace of R^n
-orthogonal complement of Row A
-Nul A^T is the orthogonal complement of Col A
Prove Nul A is a subspace of R^n - Answer-(0) Nul A is a subset of R^n because A has
n columns
(1) 0 of R^n is in Nul A, when x = 0
(2) let u and v be vector in Nul A, Au=0 and Av=0, A(u+v) = Au + Av = 0+0=0
(3) let c be any scalar, A(cu)=c(Au)=c(0)=0
Col A - Answer--Set of all linear combinations of the columns of A. If A=[a1,...,an] then
Col A = Span{a1,...,an}
-Col A = {b: b=Ax for some x in R^n}
-Col A is a subspace of R^m
-Nul A^T is the orthogonal complement of Col A
Col A is all of R^m if and only if? - Answer-the equation Ax=b has a solution for each b
in R^m
Basis - Answer-Let H be a subspace of a vector space V, an indexed set of vetors B=
{b1,...,bp} in V is a basis for H if
1) B is a linearly independent set
2) the subspace spanned by B coinsides with H, i.e., H= Span{b1,..,bp}
-If a vector space V has a basis of n vectors, then every basis of V must consist of
exactly n vectors
, each X in V can be represented as - Answer-c1b1+...+cnbn where B={b1,...,bn} is a
basis for V
Coordinates of x relative to the basis B (b-coordiantes of x) - Answer-B={b1,...,bn} is a
basis for V and x is in V. The B-coordinaets of x are the weights c1,...,cn such that x=
c1b1+...+cnbn
[x]B = [c1...cn] (Vector)
change of coordinates matrix - Answer--Pb=[b1,...,bn]
-Pb[x]B=x
-Pb^-1x=[x]B
DimV - Answer--DimV=number of vectors in a basis for V if V is finite dimensional (V is
spanned by a finite set)
-V is infinitely dimensional if V is not spanned by a finite set.
-dimension of 0 vector space is 0.
dim Nul A - Answer-number of free variables
dim Col A - Answer-number of pivot columns = rank A
row A - Answer--Set of all linear combinations of the row vectors
-subspace of R^n
-Row A = Col A^T
-If two matrices A and B are row equivalent, then their row spaces are the same. If B is
in echelon form, the nonzero entries of B form a basis for the row space of A as well as
that of B.
-Nul A is orthogonal complement of Row A
basis of Row A - Answer-nonzero rows of echelon form
rank A - Answer-dim Col A = dim Row A
Kernel - Answer--Kernel of a linear transformation T from a vector space V into a vector
space W is the set of all u in V such that T(u)= 0 (the zero vector in W)
-Nul A
