Linear Algebra Final Review Packet — Concise Proofs +
120 Practice Problems
Formula for Eigenvectors - Answers Av = (lambda) v
The symbol R - Answers All real numbers
Upper Triangular - Answers All entries below diagonal are zero
Lower Triangular - Answers All entries above diagonal are zero
Diagonal Matrix - Answers All non diagonal entries are zero
Identity Matrix - Answers 1's on the diagonal
Transpose - Answers Swap rows and columns
Involution - Answers Repeating the operation goes back to the original state
Symmetric Equation - Answers S^T=S
Trace - Answers Sum of the diagonal
Linear Combination - Answers c1v1+c2v2
Output of matrix-vector multiplication with R^n vector (remember the arrow direction) -
Answers R^m vector
,Euler Characteristic - Answers Nodes-arrows
Islands in a digraph - Answers Connected components
Cycle - Answers starts and ends on the same node
Circuit rank - Answers Minimum number of arrows that must be removed to break all cycles
Net flow - Answers weight in - weight out
Rows of incidence matrix - Answers nodes
Columns of incidence matrix - Answers arrows
Incidence matrix times weight vector - Answers All net flows
Length of [v1 v2] - Answers sqrt(v1^2 + v2^2)
||c * v|| = - Answers |c| * ||v||
Length of unit vector - Answers length 1
Normalization of v - Answers (1/||v||) * v (unit vector)
Adjoint Formula - Answers Transpose the matrix and swap to the other side of inner product
, Length and Inner Product formula - Answers <v,v>=||v||^2
Law of Cosines - Answers c^2 = a^2 + b^2 - 2abcos(theta)
Acute vectors - Answers inner product greater than 0
Obtuse vectors - Answers inner product less than 0
Orthogonal vectors - Answers dot product equals 0
Transposing AB - Answers B^TA^T
Gramian - Answers ATA
Property of ATA - Answers Symmetric
Number of pivot columns - Answers Rank
Number of non pivot columns - Answers Nullity
Rank + Nullity - Answers Number of Columns
Relationship between unpivot and pivot columns - Answers unpivot columns are linear
combinations of pivot columns
120 Practice Problems
Formula for Eigenvectors - Answers Av = (lambda) v
The symbol R - Answers All real numbers
Upper Triangular - Answers All entries below diagonal are zero
Lower Triangular - Answers All entries above diagonal are zero
Diagonal Matrix - Answers All non diagonal entries are zero
Identity Matrix - Answers 1's on the diagonal
Transpose - Answers Swap rows and columns
Involution - Answers Repeating the operation goes back to the original state
Symmetric Equation - Answers S^T=S
Trace - Answers Sum of the diagonal
Linear Combination - Answers c1v1+c2v2
Output of matrix-vector multiplication with R^n vector (remember the arrow direction) -
Answers R^m vector
,Euler Characteristic - Answers Nodes-arrows
Islands in a digraph - Answers Connected components
Cycle - Answers starts and ends on the same node
Circuit rank - Answers Minimum number of arrows that must be removed to break all cycles
Net flow - Answers weight in - weight out
Rows of incidence matrix - Answers nodes
Columns of incidence matrix - Answers arrows
Incidence matrix times weight vector - Answers All net flows
Length of [v1 v2] - Answers sqrt(v1^2 + v2^2)
||c * v|| = - Answers |c| * ||v||
Length of unit vector - Answers length 1
Normalization of v - Answers (1/||v||) * v (unit vector)
Adjoint Formula - Answers Transpose the matrix and swap to the other side of inner product
, Length and Inner Product formula - Answers <v,v>=||v||^2
Law of Cosines - Answers c^2 = a^2 + b^2 - 2abcos(theta)
Acute vectors - Answers inner product greater than 0
Obtuse vectors - Answers inner product less than 0
Orthogonal vectors - Answers dot product equals 0
Transposing AB - Answers B^TA^T
Gramian - Answers ATA
Property of ATA - Answers Symmetric
Number of pivot columns - Answers Rank
Number of non pivot columns - Answers Nullity
Rank + Nullity - Answers Number of Columns
Relationship between unpivot and pivot columns - Answers unpivot columns are linear
combinations of pivot columns
