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Definition
| Two systems of linear equations in n unknowns are equivalent provided that they have the same set of solutions |
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| Elementary Row Operations |
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Definition
1. interchange two rows
2. multiply a row by a non-zero scalar
3. add a constant multiple of one row to another. |
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Definition
1. All rows that consist entirely of zeros are grouped together at the bottom of the matrix.
2. In every non-zero row, the first non-zero entry (from left to right) is a 1.
3. if the (i+1)-st row contains non-zero entries, then the first nonzero entry is in a column to the right of the first nonzero entry in the ith row. |
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Definition
| the first non-zero entry in any row is the only non-zero entry in that column |
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Definition
| must have same numbers of rows and columns, and entries must be the same. |
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Definition
(A+B)_ij= a_ij+b_ij
Add corresponding values in two matrices. |
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Definition
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Definition
If A is a (mxn) then B must have n rows.
(AB)= sum of a_ik*b_kj from k=1 to k=n |
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Definition
| A^T= (b_ij) where b_ij=a_ji |
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Definition
if the only solution to
a1v1 +a2v2 + a3v3 +...+ apvp = 0(vector)
is the trivial solution of all a=0. |
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Definition
only solution to Ax=0 is x=0
Linked to linear independence |
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Definition
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Definition
if A is an (nxn) matrix, and there is a matrix A^-1 such that AA^-1=I
A^-1 is labeled the inverse of A |
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Definition
A is an (mxn) matrix. the null space is the set of vectors in R^n defined by
N(A)={x:Ax=0, x in R^n} |
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Definition
A is an (mxn) matrix. the range is the set of vectors in R^m defined by
R(A)={y:y=Ax for some x in R^n} |
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Definition
| S spans W (a subspace of R^n) if every vector in W can be expressed as a linear combination of the vectors in S |
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Definition
| linearly independent spanning set for a subspace. (not necessarily unique) |
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Definition
Subspace W (a subspace of R^n) has a basis B= {w1, w2, w3,...,wp} of p vectors, then W is a subspace of dimension p.
dim(W)=p |
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Definition
| Only orthogonal if each pair of distinct vectors from S is orthogonal. |
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| Orthogonal Basis and Orthonormal Basis |
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Definition
B is a basis for W. B is an orthogonal basis of W if B is an orthogonal set of vectors.
Orthonormal if all vectors are orthogonal and the length of each vector is 1. |
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Definition
V is a subspace in R^m, and W is a subspace in R^n. Let T be a function from V to W. T:V-->W. T is a linear transformation if for all u and v in V and for all scalars a
T(u+v)= T(u)+T(v)
and
T(au)=aT(u) |
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| Null Space and Range (of a transformation) |
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Definition
The null space of T is the subset of V given by
N(T)={v: v is in V and T(v)=0}
The range is the subset of W defined by
R(T)= {w: w is in W and w = T(v) for some v in V} |
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Definition
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Definition
A is an (nxn) matrix. the [(n-1)x(n-1)] matrix that results from removing the
r-th row and the s-th column from A is the minor matrix of A and is denoted M_rs |
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| Characteristic Polynomial |
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Definition
pA(t) = det(A-tI)
Hint: use an eigenvalue for t. |
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| Eigenspace and Geometric Multiplicity |
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Definition
y is used to denote an eigenvalue instead of lambda.
The null space of A-yI is denoted by E_y and is called the eigenspace of lambda.
The dimension of E_y is called the geometric multiplicity of lambda. |
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Definition
| If there is an eigenvalue of A such that the geometric multiplicity of lambda is less than the algebraic multiplicity of lambda, then A is called a defective matrix. |
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Definition
| The (nxn) matrices A and B are similar if there is a nonsingular (nxn) matrix S such that B=S^-1AS |
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