Matrix Types, Consistency, and Linear Combinations

• Matrices are defined by their dimensions as row x column.
• The number of rows and columns determines the matrix's size.

Key Matrix Types

• Augmented Matrix: An m x (n+1) matrix
• Coefficient Matrix: An m x n matrix

Equivalent Systems

• Equivalent systems of linear equations have the same solutions.
• Pivot: The leftmost nonzero entry in a matrix.
• Pivot Column: The ith column of matrix A if the reduced row echelon form (rref) of A has a pivot in column i.
• Basic Variable: Represented by xi in the ith column of A if it's a pivot column.
• Free Variable: Represented by xi in the ith column of A if it's not a pivot column.

System Consistency

• Consistent System: Has at least one solution.
• Inconsistent System: Has no solution.

Linear Combinations and Spans

• A linear combination of vectors v1, v2, ..., vn is an expression of the form w = c1 v1 + c2 v2 + ... + cn vn, where c1, c2, ..., cn are scalars (coefficients).
• Span: The span of vectors v1, v2, ..., vn in R^m is the set of all their linear combinations: span(v1, ..., vn) = {c1v1 + ... + cnvn | c1, ..., cn ∈ R}.

Linear Dependence/Independence

• Linearly Dependent Vectors: Vectors v1, ..., vn are linearly dependent if at least one vi can be written as a linear combination of the others: vi ∈ span(v1, ..., vi-1, vi+1, ..., vn).
• Linearly Independent Vectors: If not linearly dependent.
• Algebraic Definition of Linear Dependence: Vectors v1, ..., vn are linearly dependent if there exists a nontrivial solution (not all scalars equal to zero) to the equation c1v1 + ... + cnvn = 0.
• Vectors are linearly independent if the only solution is the trivial one (all scalars are zero).

Subspaces and Vector Spaces

• Subspace: A subspace V of R^n is a nonempty subset of R^n that satisfies:
  • Closure under vector addition: If u, v ∈ V, then u + v ∈ V.
  • Closure under scalar multiplication: If c ∈ R and u ∈ V, then cu ∈ V.
• Vector Space: Any set V satisfying the subspace properties.

Spanning/Generating Sets

• A vector space V is spanned/generated by vectors v1, ..., vn if every vector in V can be written as a linear combination of v1, ..., vn: V = span(v1, ..., vn).
• The set {v1, ..., vn} is called a spanning/generating set for V.

Basis

• A basis B of a vector subspace V of R^n is a linearly independent generating set.
• The vectors in B span V and are linearly independent.

Dimension

• The dimension of a vector subspace V is the number of vectors in any basis for V.

Column Space and Null Space

• Given an m x n matrix A with columns v1, ..., vn:
  • Column Space: The subspace of R^m spanned by the columns of A: Col(A) = span(v1, ..., vn).
  • Null Space: The subspace of R^n containing all vectors x such that Ax = 0: Nul(A) = {x ∈ R^n | Ax = 0}.

Rank and Nullity

• Rank: The dimension of the column space of A (number of pivot columns).
• Nullity: The dimension of the null space of A (number of non-pivot columns).

Transpose of a Matrix

• Given an m x n matrix A, the transpose AT is an n x m matrix where the columns of AT are equal to the rows of A.

Row Space

• The row space of a matrix A, Row(A), is the subspace of R^n spanned by the row vectors of A.

Linear Transformations

• A function F: R^n → R^m is a linear transformation if it satisfies:
  • F(x + y) = F(x) + F(y) for all vectors x, y ∈ R^n.
  • F(cx) = cF(x) for all vectors x ∈ R^n and scalars c ∈ R.
• Matrix Transformation: The function TA: R^n → R^m defined by TA(x) = Ax, where A is an m x n matrix. All matrix transformations are linear transformations.

Defining Matrix

• If F: R^n → R^m is a linear transformation, then it can be represented/defined by an m x n matrix AF, which is called the defining matrix of F.

Matrix Product

• Given an m x k matrix A and a k x n matrix B, the matrix product C = AB is an m x n matrix such that TA TB = TC.

Kernel and Image

• Given a linear transformation F: R^n → R^m:
  • Kernel: The set of all vectors x in R^n such that F(x) = 0: ker(F) = {x ∈ R^n | F(x) = 0}.
  • Image: The set of all vectors y in R^m such that y = F(x) for some x in R^n: im(F) = {y ∈ R^m | y = F(x) for some x ∈ R^n}.

Injectivity and Surjectivity

• Given sets X and Y and a function f: X → Y:
  • Injective (one-to-one): For every y ∈ Y, there is at most one x ∈ X such that f(x) = y.
  • Surjective (onto): For every y ∈ R^m, there is at least one x ∈ R^n such that F(x) = y.
  • Bijective: Both one-to-one and onto.

Isomorphism

• Given a subspace V of R^n and a subspace W of R^m, an isomorphism is any linear bijective map F: V → W.

Identity Matrix

• The defining matrix of the identity transformation Iden: R^n → R^n, where Iden(x) = x.

Inverse Matrix

• Given an n x n matrix A:
  • Geometric Definition: The inverse matrix of A is the defining matrix of the inverse transformation TA-1.
  • Algebraic Definition: The inverse of A is an n x n matrix B satisfying AB = BA = In, where In is the n x n identity matrix.

Rouche-Capelli Theorem

• For a system of linear equations with augmented matrix A and coefficient matrix C:
  • The system is inconsistent if the last column of rref(A) has a pivot.
  • The system has exactly one solution if the last column of rref(A) does not have a pivot and every column of rref(C) has a pivot.
  • The system has infinitely many solutions if the last column of rref(A) does not have a pivot, and rref(C) has a column without a pivot.

Linear Dependence Definitions

• Two definitions of linear dependence agree. Explain how the geometric definition of linear dependence implies the algebraic one. Explain how the algebraic definition of linear dependence implies the geometric one:

Theorems

• A subset V is a vector subspace of $R^n$ if and only if there exists vectors $v_1, ... , v_m$ so that $V = Span(v_1, ... , v_m)$. Let A be an m x n matrix of the form $A = (v_1 v_2 ... vm)$ where the $v_i$ are vectors in $R^n$ and suppose that $rref(A) = (v_1 v_2 ... vm)$. If the column $vm$ of $rref(A)$ does not have a pivot, then $Span(v_1, ... , v_m) = Span(v_1, ... , v_{m-1}).$

  • In general, we can remove any column of A that is not a pivot column and not change the span of its column vectors.

• Let A be an m x n matrix with r pivot columns. Then rank(A) = r and nullity(A) = n-r. That is, the rank of A is equal to the number of pivot columns of A, and the nullity of A is the number of non-pivot columns of A.

• The solution set to any homogeneous system of equations is a vector space. Furthermore, if the system has coefficient matrix A, then the solution set is equal to Nul(A).

• Every linear transformation is a matrix transformation. In particular, if $F:R^n \implies R^m$ is linear, then $F = T_A$ where $A = (F(e_1) ... F(e_n ))$.

Injectivity and Surjectivity Theorem

• Let F : $R^n → R^m$ be a linear transformation with defining matrix $A_F$. Then,
• F is injective if and only if every column in rref($A_F$) has a pivot.
• F is surjective if and only if every row in rref($A_F$) has a pivot.