Math 307, Section D - Study Guide: Exam #2
1. Section 2.8: Subspaces of Rn
• Recall: A subspace of Rn is any set H ∈ Rn that satisfies:
(a) The zero vector is in H.
(b) For each u, v ∈ H, the sum u + v ∈ H.
(c) For each u ∈ H and each scalar c ∈ R, the vector cu ∈ H.
• Recall: A basis for a subspace H ⊆ Rn is a linearly independent set in H that spans H.
• If A = [a1 a2 . . . an ] ∈ Rm×n then the column space of A, denoted Col(A), is the
set Span{a1 , a2 , . . . , an }. A basis for Col(A) is comprosed of the linearly independent
columns of A. Make sure you know how to find the basis determine if a vector b ∈ Rm ∈
Span{a1 , a2 , . . . , an }.
• The null space of a matrix A ∈ Rm×n , denoted N ul(A), is a subspace of Rn that is
composed of the set of all solutions to the homogenous equation Ax = 0, where x ∈ Rn .
Know how to determine a basis for this subspace.
• A matrix A ∈ Rm×n can be used to define a linear transformation T : Rn → Rm defined
by T (x) = Ax. Make sure you understand how the null space and the column space can
be used to partition the the domain, Rn , and the codomain, Rm .
2. Section 2.9: Dimension and Rank
• Recall: The dimension of a nonzero subspace H, denoted dim(H), is the number of
vectors in any basis for H. By definition, dim ({0}), the dimension of the zero subspace
is zero.
• Recall: The rank of a matrix A ∈ Rm×n , denoted rank(A), is the dimension of column
space of A.
• Recall: Given a basis B = {b1 , b2 , . . . , bp } for a subset H, then for each x ∈ H the
coordinates of x relative to the basis B are the weights c1 , c2 , . . . , cp such that
x = c1 b1 + c2 b2 + . . . + cn bn , and the vector
 
c1
 c2 
 
[x]B =  .  ∈ Rp
 .. 
cn
is called the corrdinate vector of x relative to B.
• Given a basis B for a subspace H ⊆ Rn and a vector x ∈ Rn , now how determine if
x ∈ H. Also, make sure that you know how to determine the constants c1 , c2 , . . . , cn
that defines the linear combination of the basis vectors that yields x.
• Finally, make sure that you understand and can apply the Theorems on page 156.
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3. Section 3.1: Introduction to Determinants
• Recall: Given matrix A ∈ Rn×n , the determinant of A, denoted Det(A), can be computed
(a) through a cofactor expansion accross the ith row:
Det(A) = (−1)i+1 ai1 Det(Ai1 ) + (−1)i+2 ai2 Det(Ai2 ) + . . . + (−1)i+n ain Det(Ain )
(b) through a cofactor expansion down the j th column:
Det(A) = (−1)1+j a1j Det(A1j ) + (−1)2+j a2j Det(A2j ) + . . . + (−1)n+j anj Det(Anj )
where aij denotes the element of A located in row i column j and Aij denotes the matrix
obtained by crossing out row i and column j from A.
• Given a matrix A ∈ Rm×n with column vectors a1 , a2 , . . . , an and the vector b ∈ Rm ,
the matrix equation Ax = b has a solution iff b ∈ Span{a1 , a2 , . . . , an }.
• Make sure you know how to recursively apply the above method to determine the determinant of large matrices.
• Also, make sure you now how to calculate the determinant of a diagonal matrix quickly.
4. Section 3.2: Properties of Determinants
• Recall: Given matrices A, B ∈ Rn×n ,
(a) If a multiple of one row of A is added to another row of A to produce B, then
Det(B) = Det(A)
(b) If two rows of A are interchanged to produce B, then
Det(B) = −Det(A)
(c) If one row of A is multiplied by k to produce B, then
Det(B) = k · Det(A)
• Make sure that you know how to apply the above properties on a sequence of row
operations. If you are given the value for the determinant of an initial matrix, you
should be able to determine the value of the determinant for the matrix that results
after a sequence of of row operations.
• Recall: Given a matrix A, B ∈ Rn×n ,
(a) A is invertible iff Det(A) 6= 0.
(b) Det(AT ) = Det(A). (This is clear due that the determinant can be calculated using
a cofactor expansion down a column OR accross a row.)
(c) Det(AB) = [Det(A)] · [Det(B)].
• Make sure that you understand how to apply the above properties. They make it much
easier to determine the value for the determinant of a matrix quickly.
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5. Section 3.3: Cramer’s Rule, Volume, and Linear Transformations
• Recall: Given an invertible matrix A ∈ Rn×n , for any b ∈ Rn the unique solution x of
Det(Ai (b))
i = 1, 2, . . . , n where
the linear system Ax = b has entries given by xi =
Det(A)
Ai (b) denotes the matrix obtained by replacing the ith column of A with the vector b.
• NOTE: The above theorem only demonstrates how to determine an individual entry in
the solution vector. Thus, the above idea has to be repeated n times in order to obtain
the entire solution vector.
• Recall: Given an invertible matrix A ∈ Rn×n , then the adjugate of A is defined as:


C11 C12 . . . C1n
 C21 C22 . . . C2n 


Adj(A) =  .
..
.. 
.
.
.
 .
.
.
. 
Cn1 Cn2 . . . Cnn
where Cji = Det (Ai (ej )) = (−1)i+j Det(Aj i).
• Recall: Given an invertible matrix A ∈ Rn×n , then
A−1 =
1
Adj(A)
Det(A)
where the Adj(A) is as defined above.
• Make sure that you also understand how the determinant can be used to calculate the
area/volume of shape after it has been modified by a transformation.
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