Consistent | If it has at least one solution. |
Row equivalent | If a sequence of row operations transforms one matrix into the other. |
Unique solution | If and only if there are no free variables |
Homogeneous | Linear systems of the form Ax=0 |
Inhomogeneous | Linear systems of the form Ax=b where b=0 |
Trivial solution | The solution is the zero vector |
Linearly independent | If no vector can be made from other vectors |
Row operations | Addition, Interchange, Scaling |
Pivot position | A leading 1 in the RREF of A |
Pivot column | Is a column of A that contains a pivot position |
Domain | T:Rn →Rm; Rn is the domain of T |
Codomain | T:Rn →Rm; Rm is the codomain of T |
Image | The vector T(x) is the image of x under T |
Range | The set of all possible images T(x) or simply the span of A |
Standard vectors | The column of the identity matrix (think [10] and [01]) |
Onto | All the elements in the codomain are mapped to. (A spans the entire codomain), Every row is pivotal |
One-To-One | Each mapping is unique (2 vectors can NOT map to the same vector), Every column is pivotal |
Transpose | The matrix whose columns are the rows of A |
Invertible | A∈Rn×n is invertible if there is a C∈Rn×n such that: AC=CA=In |
Elementary Matrix | Differs from In by one row operation. |
Singular | A matrix that is not invertible (A−1 DNE) |
Subset | A subset of Rn any collection of vectors that are in Rn |
Subspace | If H∈Rn, for c∈R and u,v∈H, cu∈H and u+v∈H must be true if H is a subspace. |
Column Space | This is a subspace spanned by the column of A. |
Null Space | This is a subspace spanned by all x such that Ax=0. |
Basis | This is a set of linearly independent vectors in H that spans H assuming H is a subspace. |
Coordinate Vector | These are the vectors that are used to describe the coordinate systems. |
Coordinates | These are the weights of the coordinate vector used to describe the point. |
Dimension | This is the number of vectors in a basis of H. |
Cardinality | Same thing as Dimension |
Rank | Rank(A)=dim(Col(A))=no of pivot columns |
Determinant | It is the scaling factor that tells us how a transformation will change the area or volume of a region. |
Probability Vector | A vector with non-negative elements that sum to 1 |
Stochastic Matrix | Square matrix, P, whose columns are probability vectors. |
Markov Chain | The sequence: xk+1=Pxk (0≤k) |
Steady-State Vector | Is the a probability vector such that Pq=q |
Regular Stochastic Matrix | If there is some k such that Pk only contains positive entries. |
Trace | The sum of the elements of the main diagonal. |
Characteristic Polynomial | det(A−λI) |
Characteristic Equation | det(A−λI)=0 |
Multiplicity | The number of times that its associated factor appears in the polynomial. |
Algebraic Multiplicity | Multiplicity of the characteristic polynomial. |
Geometric Multiplicities | Dimensions of Null(A−λI) |
Similar Matrices | A and B are similar if there is a P such that A=PBP−1. |
Diagonal Matrices | If the only non-zero elements, if any, are on the main diagonal. |
Diagonalizable | If A is similar to diagonal matrix D (A=PDP−1) |
Unit Vector | When the length of a vector is 1 |
Orthogonal | If u⋅v=0, then u,v are Orthogonal |
Row space | the space spanned by the rows of matrix A |
Orthogonal Sets | If for set {u1,…,vn} for j=k, uj⊥uk. |
Orthonormal Columns | An m×n,m≥n matrix has orthonormal columns ⟺UTU=In |
Symmetric | A matrix is symmetric when A=AT |
Spectrum | The set of eigenvalues of a matrix |
positive definite | If Q>0 for all x=0 |
negative definite | If Q<0 for all x=0 |
positive semidefinite | If Q≥0 for all x |
negative semidefinite | If Q≤0 for all x |
Indefinite | If Q takes on positive and negative values for x=0 |
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