LINEAR ALGEBRA EXAM QUESTIONS
WITH COMPLETE SOLUTIONS
orthonormal - Answer-vectors are perpendicular and have unit length 1
orthogonal - Answer-perpendicular vectors, dot product is zero
normalize a vector - Answer-divide each entry by its length
linear combination - Answer-addition of vectors with their scalar multiples
identity matrix - Answer-ones along diagonal, zeros everywhere else
(A^-1)A = - Answer-identity matrix
A(I)= - Answer-A
Inverse (A^-1) - Answer-what you need to multiply A by to return the identity matrix
matrix multiplication properties - Answer-associative: A(BC)=(AB)C
left distributive: A(B+C) = AB + AC
right distributive: (A+B)C = AC + BC
not commutative: AB != BA
determinant - Answer-unique scalar value calculated from a square matrix used to
calculate the inverse
for 2x2: ad-bc (diagonals)
zero if linearly dependent, zero if invertible
elimination matrices - Answer-matrix representation of gauss-jordan elimination to solve
a linear system of equations.
lower triangular matrix which contains the negated multiplier (m=c/p) for each variable
permutation matrices - Answer-used to permute the order of rows (equations) to better
perform elimination and thus solve the system
row echelon form - Answer-1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row
above it.
3. All entries in a column below a leading entry are zeros.
, reduced row echelon from - Answer-all pivots values from row echelon form are
reduced to ones, provides simplest/smallest possible solution to system of equations
multiplier (m=) - Answer-coefficient of x1 n eq2 (c) / coefficient of x1 in eq1 (p)
vector spaces - Answer-given vector u is an element in the vector space, it's linear
combinations also lie within the vector space.
common ones are R2, R3, etc.
closed under vector addition and scalar multiplication
properties:
span - Answer-the set of all possible linear combinations for a given vector space
linearly independent - Answer-vector u is not a linear combination of vector v
basis vector - Answer-linearly independent and span the vector space.
if in 2d space there would be 2 linearly independent vectors which spanned the space, if
in 3d space there would be 3 linearly dependent vectors which spanned the space, and
so on
subspace - Answer-subset of a vector space.
given u and v are vectors in the subspace, then u+v exists in the subspace as well as
any scalar multiple of u or v.
column space - Answer-set of all linear combinations of the columns of a matrix, also
known as b in Ax=b
null space - Answer-set of all possible solutions when Ax=0, must always contain the
zero vector
pivot column - Answer-first nonzero entry of each row when matrix in row echelon form,
the linearly independent variables
rank - Answer-=number of pivot columns in row echelon form
=number unknowns/columns - number equations/rows in r.e.f.
rank=n then called full rank and has 1 unique solution
free variables - Answer-columns without pivots
factorization - Answer-breaking A into multiple matrices that when multiplied together
return A
WITH COMPLETE SOLUTIONS
orthonormal - Answer-vectors are perpendicular and have unit length 1
orthogonal - Answer-perpendicular vectors, dot product is zero
normalize a vector - Answer-divide each entry by its length
linear combination - Answer-addition of vectors with their scalar multiples
identity matrix - Answer-ones along diagonal, zeros everywhere else
(A^-1)A = - Answer-identity matrix
A(I)= - Answer-A
Inverse (A^-1) - Answer-what you need to multiply A by to return the identity matrix
matrix multiplication properties - Answer-associative: A(BC)=(AB)C
left distributive: A(B+C) = AB + AC
right distributive: (A+B)C = AC + BC
not commutative: AB != BA
determinant - Answer-unique scalar value calculated from a square matrix used to
calculate the inverse
for 2x2: ad-bc (diagonals)
zero if linearly dependent, zero if invertible
elimination matrices - Answer-matrix representation of gauss-jordan elimination to solve
a linear system of equations.
lower triangular matrix which contains the negated multiplier (m=c/p) for each variable
permutation matrices - Answer-used to permute the order of rows (equations) to better
perform elimination and thus solve the system
row echelon form - Answer-1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row
above it.
3. All entries in a column below a leading entry are zeros.
, reduced row echelon from - Answer-all pivots values from row echelon form are
reduced to ones, provides simplest/smallest possible solution to system of equations
multiplier (m=) - Answer-coefficient of x1 n eq2 (c) / coefficient of x1 in eq1 (p)
vector spaces - Answer-given vector u is an element in the vector space, it's linear
combinations also lie within the vector space.
common ones are R2, R3, etc.
closed under vector addition and scalar multiplication
properties:
span - Answer-the set of all possible linear combinations for a given vector space
linearly independent - Answer-vector u is not a linear combination of vector v
basis vector - Answer-linearly independent and span the vector space.
if in 2d space there would be 2 linearly independent vectors which spanned the space, if
in 3d space there would be 3 linearly dependent vectors which spanned the space, and
so on
subspace - Answer-subset of a vector space.
given u and v are vectors in the subspace, then u+v exists in the subspace as well as
any scalar multiple of u or v.
column space - Answer-set of all linear combinations of the columns of a matrix, also
known as b in Ax=b
null space - Answer-set of all possible solutions when Ax=0, must always contain the
zero vector
pivot column - Answer-first nonzero entry of each row when matrix in row echelon form,
the linearly independent variables
rank - Answer-=number of pivot columns in row echelon form
=number unknowns/columns - number equations/rows in r.e.f.
rank=n then called full rank and has 1 unique solution
free variables - Answer-columns without pivots
factorization - Answer-breaking A into multiple matrices that when multiplied together
return A
