MATH 237 INTRO TO LINEAR ALGEBRA
FINAL EXAM QUESTIONS WITH
CORRECT ANSWERS
For an mxn matrix, m = _______ and n = ______ - Answer-rows / columns
EROs are - Answer-Elementary Row Operations
Two matrices are row-equivalent if they differ by - Answer-a finite series of EROs
An upper triangular matrix is one where - Answer-everything below the diagonal is 0
A pivot in a matrix is - Answer-a 1 in a column full of 0's
Inconsistent System - Answer-a linear system with no solution
Consistent System - Answer-a linear system with at least one solution
Reduced Row Echelon Form (RREF) - Answer-also known as Gaussian-Jordan
Elimination. It's a matrix where the diagonal is 1's, and everything else is 0's
Free Variable - Answer-variable corresponding to columns that do not have a pivot
Particular Solution - Answer-the constants after the = sign in a linear equation
Homogeneous Solution - Answer-the coefficients of the variables in a linear equation
N-dimensional vector - Answer-aka an N-vector is a sequence of N real numbers.
Magnitude - Answer-length of a vector
Dot product of two vectors - Answer-u1×v1+...+un×vn
Cos(theta) Theorem - Answer-if theta is the angle between vectors u and v, then u°v= ||
u|| × ||v|| × cos(theta)
This also means cos(theta) = (u°v)/(||u|| × ||v||)
Orthogonal Theorem - Answer-when vectors u and v are orthogonal (aka
perpendicular), u is orthogonal to v, if and only if u°v=0.
Important properties of dot product - Answer-First: the symmetric property - u°v = v°u
Second: the distributive property - u°(v+w) = u°v + u°w
Third: bilinear property - u°(2v+3w) = 2u°v + 3u°w
, Forth: positive definite property - u°u = ||u||² > 0, ergo if u°u=0, then u=0 vector (entirely
full of zeroes)
Cauchy-Schwarz inequality - Answer-For any non zero vectors u and v, the absolute
value of their dot product must be less than or equal to the product of their lengths: |u°v|
< ||u|| × ||v||
Triangle inequality - Answer-For any u and v EIR^n: ||u+v|| < ||u||+||v||
Commutative law of addition - Answer-A+B = B+A
Associative law of addition - Answer-A+(B+C) = (A+B)+C
Distributive law of scalars - Answer-α(A+B) = αA + αA
Transpose of a matrix - Answer-if A is an mxn matrix, then its transpose, At, is the n x m
matrix whose rows are the columns of A, and the columns are the rows of A.
Transpose Operations - Answer-(AT)T = A
(A+B)T = AT + BT
(2A)T = 2AT
Symmetric matrix - Answer-a square matrix where AT = A.
Trace of a matrix - Answer-the sum of elements on the main diagonal of A
Properties of the trace - Answer-tr(A+B) = tr(A) + tr(B)
tr(2A) = 2tr(A)
tr(A) = tr(AT)
tr(AB) = tr(BA) Note this is not dot product.
Matrix multiplication properties - Answer-Where A, B, and C are matrices of appropriate
size
Associativity: A(BC) = (AB)C
Distributive: A(B+C) = AB + AC
Distributive: (A+B)C = AC+BC
Scalar: 2(AB) = (2A)B = A(2B)
Inverse Matrix - Answer-Let A and B be n x n matrices. Then B is an inverse of A if the
following are true:
AB = BA = In
If B is an inverse, we want to denote it "B as A-1" or (B=A-1)
Then (A=B-1) is also true.
Non-Invertible/Singular Matrix - Answer-The square matrix A is singular if A-1 does not
exist.
FINAL EXAM QUESTIONS WITH
CORRECT ANSWERS
For an mxn matrix, m = _______ and n = ______ - Answer-rows / columns
EROs are - Answer-Elementary Row Operations
Two matrices are row-equivalent if they differ by - Answer-a finite series of EROs
An upper triangular matrix is one where - Answer-everything below the diagonal is 0
A pivot in a matrix is - Answer-a 1 in a column full of 0's
Inconsistent System - Answer-a linear system with no solution
Consistent System - Answer-a linear system with at least one solution
Reduced Row Echelon Form (RREF) - Answer-also known as Gaussian-Jordan
Elimination. It's a matrix where the diagonal is 1's, and everything else is 0's
Free Variable - Answer-variable corresponding to columns that do not have a pivot
Particular Solution - Answer-the constants after the = sign in a linear equation
Homogeneous Solution - Answer-the coefficients of the variables in a linear equation
N-dimensional vector - Answer-aka an N-vector is a sequence of N real numbers.
Magnitude - Answer-length of a vector
Dot product of two vectors - Answer-u1×v1+...+un×vn
Cos(theta) Theorem - Answer-if theta is the angle between vectors u and v, then u°v= ||
u|| × ||v|| × cos(theta)
This also means cos(theta) = (u°v)/(||u|| × ||v||)
Orthogonal Theorem - Answer-when vectors u and v are orthogonal (aka
perpendicular), u is orthogonal to v, if and only if u°v=0.
Important properties of dot product - Answer-First: the symmetric property - u°v = v°u
Second: the distributive property - u°(v+w) = u°v + u°w
Third: bilinear property - u°(2v+3w) = 2u°v + 3u°w
, Forth: positive definite property - u°u = ||u||² > 0, ergo if u°u=0, then u=0 vector (entirely
full of zeroes)
Cauchy-Schwarz inequality - Answer-For any non zero vectors u and v, the absolute
value of their dot product must be less than or equal to the product of their lengths: |u°v|
< ||u|| × ||v||
Triangle inequality - Answer-For any u and v EIR^n: ||u+v|| < ||u||+||v||
Commutative law of addition - Answer-A+B = B+A
Associative law of addition - Answer-A+(B+C) = (A+B)+C
Distributive law of scalars - Answer-α(A+B) = αA + αA
Transpose of a matrix - Answer-if A is an mxn matrix, then its transpose, At, is the n x m
matrix whose rows are the columns of A, and the columns are the rows of A.
Transpose Operations - Answer-(AT)T = A
(A+B)T = AT + BT
(2A)T = 2AT
Symmetric matrix - Answer-a square matrix where AT = A.
Trace of a matrix - Answer-the sum of elements on the main diagonal of A
Properties of the trace - Answer-tr(A+B) = tr(A) + tr(B)
tr(2A) = 2tr(A)
tr(A) = tr(AT)
tr(AB) = tr(BA) Note this is not dot product.
Matrix multiplication properties - Answer-Where A, B, and C are matrices of appropriate
size
Associativity: A(BC) = (AB)C
Distributive: A(B+C) = AB + AC
Distributive: (A+B)C = AC+BC
Scalar: 2(AB) = (2A)B = A(2B)
Inverse Matrix - Answer-Let A and B be n x n matrices. Then B is an inverse of A if the
following are true:
AB = BA = In
If B is an inverse, we want to denote it "B as A-1" or (B=A-1)
Then (A=B-1) is also true.
Non-Invertible/Singular Matrix - Answer-The square matrix A is singular if A-1 does not
exist.
