LINEAR ALGEBRA EXAM 2 REVIEW
QUESTIONS AND ANSWERS
Matrix Addition - Answer-if A and B are mxn matrices, then the sum A + B is the mxn
matrix whose columns are the sums of corresponding columns in A and B.
Ex: [4 -3] + [1 4] = [5 1]
Matrix Scaling - Answer-if r is a scalar and A is a matrix, then the scalar multiple rA is
the matrix whose columns are r times the corresponding columns in A.
Ex: 3[2 0 5] = [6 0 15]
Matrix Multiplication - Answer-if A is an mxn matrix, and if B is an nxp matrix with
columns b, b2, ..., bp, then the product AB is the mxp matrix whose columns are Ab1,
Ab2, ... ,Abp.
(columns of first matrix being multiplied must be equal to the rows of the second matrix
being multiplied). Warning= BA may not necessarily equal AB
Inverse of a Matrix - Answer-suppose A is an nxn matrix (only considering square
matrixes here). We say that A is invertible if there exists an nxn matrix x such that AX =
XA = I
if A is invertible and X is the matrix satisfying the equation above, then we say that X is
the inverse of A and use the notation A^-1 = X.
Find inverse steps - Answer-1) augment A with identity matrix [ A | Identity ]
2) reduce to row echelon form
- if A is not row equivalent with the identity matrix, then not invertible
- if A is equivalent to the identity matrix then it IS invertible --> operations applied to I
will yield A^-1 on the right side.
[ A | Identity ] ~ [ Identity | A^-1 ]
Vector Spaces - Answer-we can apply ideas from R^n (span, linear independence,
linear transformations) in any setting where we can take linear combinations
> a vector space is a non-empty set of objects called vectors V on which are defined
two operations, addition and multiplication by scalars (real #s) subject to the axioms
below. The axioms must hold for all vectors.
1) The sum u+v must be an element of V
2) u+v = v+u
3) u + (v+w)= (u+v)+w
4) there is a zero vector 0 in V satisfying u+0=u
, 5) there is a vector -u in V such that u+ (-u) = 0
6) the scalar multiple of cu is in V
7) c(u+v)=cu+cv
8) (c+d)u= cu+du
9) c(du)=(cd)u
10) 1u = u
Null Space - Answer-the null space of an mxn matrix A, denoted NulA, is a set of all
solutions x to the homogeneous equation Ax=0
Null Space Theorem - Answer-for any mxn matrix A, NulA is a subspace of r^n
def. A subset W of a vector space is a subspace if:
1) 0 is in W
2) if u and v are in W, then u+v is in W
3) if c is a scalar and u is in W, then cu is in W
The unique representation theory - Answer-let B = {b1,...,bn} be a basis for vector space
V. Then for each x in V there exists a unique set of scalars c1,..., cn such that
x=c1b1+...+cnbn
definition: Suppose B = {b1,...,bn} is a basis for V and x is in V . The coordinates of x
relative to the basis B are the weights c1,...cn such that x=c1b1+...+cnbn.
If c1,...,cm are the B-coordinates of x, then the vector in R^n
[x]subB = [ c1
c2
...
cn ]
is the coordinate vector of x (relative to B)
Rank - Answer-the rank of matrix A is the dimension of the column space of A
Rank A= dimColA
In general, rank A = dimColA
= # of pivots in A
= # of basic variables in Ax = 0
In general, dimNulA = # of free variables in Ax=0
= # of non-pivot columns
in an mxn matrix:
rank A(# of basic var.) + dimNulA(# of free var) = n(total # var.) } rank theorem
Ex: If the null space of an 11x7 matrix is 5-dimensional, then what is the rank of A?
formula: rank A + dimNulA = n
rank A + 5 = 7
rank A = 2
QUESTIONS AND ANSWERS
Matrix Addition - Answer-if A and B are mxn matrices, then the sum A + B is the mxn
matrix whose columns are the sums of corresponding columns in A and B.
Ex: [4 -3] + [1 4] = [5 1]
Matrix Scaling - Answer-if r is a scalar and A is a matrix, then the scalar multiple rA is
the matrix whose columns are r times the corresponding columns in A.
Ex: 3[2 0 5] = [6 0 15]
Matrix Multiplication - Answer-if A is an mxn matrix, and if B is an nxp matrix with
columns b, b2, ..., bp, then the product AB is the mxp matrix whose columns are Ab1,
Ab2, ... ,Abp.
(columns of first matrix being multiplied must be equal to the rows of the second matrix
being multiplied). Warning= BA may not necessarily equal AB
Inverse of a Matrix - Answer-suppose A is an nxn matrix (only considering square
matrixes here). We say that A is invertible if there exists an nxn matrix x such that AX =
XA = I
if A is invertible and X is the matrix satisfying the equation above, then we say that X is
the inverse of A and use the notation A^-1 = X.
Find inverse steps - Answer-1) augment A with identity matrix [ A | Identity ]
2) reduce to row echelon form
- if A is not row equivalent with the identity matrix, then not invertible
- if A is equivalent to the identity matrix then it IS invertible --> operations applied to I
will yield A^-1 on the right side.
[ A | Identity ] ~ [ Identity | A^-1 ]
Vector Spaces - Answer-we can apply ideas from R^n (span, linear independence,
linear transformations) in any setting where we can take linear combinations
> a vector space is a non-empty set of objects called vectors V on which are defined
two operations, addition and multiplication by scalars (real #s) subject to the axioms
below. The axioms must hold for all vectors.
1) The sum u+v must be an element of V
2) u+v = v+u
3) u + (v+w)= (u+v)+w
4) there is a zero vector 0 in V satisfying u+0=u
, 5) there is a vector -u in V such that u+ (-u) = 0
6) the scalar multiple of cu is in V
7) c(u+v)=cu+cv
8) (c+d)u= cu+du
9) c(du)=(cd)u
10) 1u = u
Null Space - Answer-the null space of an mxn matrix A, denoted NulA, is a set of all
solutions x to the homogeneous equation Ax=0
Null Space Theorem - Answer-for any mxn matrix A, NulA is a subspace of r^n
def. A subset W of a vector space is a subspace if:
1) 0 is in W
2) if u and v are in W, then u+v is in W
3) if c is a scalar and u is in W, then cu is in W
The unique representation theory - Answer-let B = {b1,...,bn} be a basis for vector space
V. Then for each x in V there exists a unique set of scalars c1,..., cn such that
x=c1b1+...+cnbn
definition: Suppose B = {b1,...,bn} is a basis for V and x is in V . The coordinates of x
relative to the basis B are the weights c1,...cn such that x=c1b1+...+cnbn.
If c1,...,cm are the B-coordinates of x, then the vector in R^n
[x]subB = [ c1
c2
...
cn ]
is the coordinate vector of x (relative to B)
Rank - Answer-the rank of matrix A is the dimension of the column space of A
Rank A= dimColA
In general, rank A = dimColA
= # of pivots in A
= # of basic variables in Ax = 0
In general, dimNulA = # of free variables in Ax=0
= # of non-pivot columns
in an mxn matrix:
rank A(# of basic var.) + dimNulA(# of free var) = n(total # var.) } rank theorem
Ex: If the null space of an 11x7 matrix is 5-dimensional, then what is the rank of A?
formula: rank A + dimNulA = n
rank A + 5 = 7
rank A = 2
