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**1. Systems of Linear Equations: Algebra
- Systems of Linear Equations: Geometry** 2.1. Span of columns and Solution set
A = m x n Lives in Dimension
Span of columns of A Rm^ Number of pivots
Solution set of Ax = 0 Rn^ Number of free variables
2.2. Linear Independence and Dependence
- Check for linear independence
- Row reduce for Ax = 0 โ verify that x = y = z = 0 OR pivot in every column
- Wide matrix โ cannot be linearly independent
2.3. Subspaces
- Notation: V = { x,y in R^2 | x + y = 1}
- Check for subspace:
- Contain 0 : sub. x = y = z = .. = 0 into subspace equation
โ
- Closed under addition: Let u = (a b) and v = (c d) => u + v = (a+c, b+d)
- Closed under multiplication: Let u = (a b) โ cu = c(a b)
- Common types of subspaces:
- Col(A): spans of columns of A
- Null(A): solution set of Ax = 0
- Find A from Col(A) and Nul(A) โ Eg. Col(A) = (x y), Nul(A) = (3 1) โ x 1 = 3x 2 โ x 1 - 3x 2 = 0 โ A = (x -3x y -3y)
- Subspace contains the vector = Span of that vector
- Not subspaces:
- y = |x|: (1 1) and (-1 -1)
- x^2 + y^2 โค 1: (1 0) and (5 0)
- y โฅ 2x
- xy = 0: (1 0) and (0 1)
2.4. Basis and Dimension
- Dim(V) = number of vectors in any basis of V
- If V = Rn^ (n vectors) โ any basis of V has n vectors in it
- If V = Col(A), dim(V) = dim(ColA) = rank(A) = number of pivot columns
- If V = Nul(A), dim(V) = dim(NullA) = nullity(A) = number of free variables
- Verify basis: i) Vectors are in V ii) Vectors span V iii) Vectors are linearly independent OR if dimV = m, any m vectors either span V OR are linearly independent
- Compute basis for a subspace:
- Basis of Col(A): Row reduction โ Original pivot columns โ Another basis of the same span / column space: any non-zero linear combination of vectors in the basis
- Basis of a span: find basis of Col(A)
- Basis of Nul(A): Parametric form of general solution โ Another vector โ 0 and in Nul(A): any non-zero linear combination of vectors in the basis
- Basis of a subspace: โ Convert equation to the form Ax = 0 => V = Nul(a b c) โ Find parametric form of general solution
2.5. Rank Theorem
- rank(A) + nullity(A) = number of columns of A 3. Linear Transformations and Matrix Algebra
3.1. Matrix Transformation
- Let A be an mรn matrix, and let T(x) = Ax be the associated matrix transformation.
- Domain(T) = Rn
- Codomain(T) = Rm
- Range(T) = Col(A) = n-D object in Rm
- Dim(T) = dim(Range(T)) = dim(Col(A)) = rank(A) = number of pivot columns
- T : Rn^ โ Rm^ โ A(x) = m x n
3.2. One-to-one and Onto Transformation
Non-examples of Tranformation
4. Determinants
4.1. Cofactor Expansion Let A be an n ร n matrix with entries aij
det(A) = with ๐ = 1
๐ โ ๐๐๐๐ถ๐๐ ๐ถ๐๐ = (โ 1) ๐+๐ ๐๐๐ก(๐ด๐๐)
5. Eigenvalues and Eigenvectors
5.1. Definition
- An eigenvector of A is a nonzero vector v in R n such that Av = ฮปv, for some scalar ฮป
- An eigenvalue of A is a scalar ฮป such that the equation Av = ฮปv has a nontrivial solution
- ฮป = 1 โ (A - I) has a non-trivial solution โ A - I is not invertible
- Dim(ฮป-eigenspace) = number of free variable
Matrix ฮป
Reflection 1, -
Projection 0, 1
Identity 1
Shear 1
Rotation by theta < 180 No real ฮป
Rotation by 180 aka reflection 1 (counterclockwise) OR -1 (clockwise)
5.2. Characteristic Polynomial
- f(ฮป) = ฮป^2 - Tr(A)ฮป + det(A)
- Trace(A) = sum of diagonal entries
5.3. Diagonalization
- Diagonalizable matrices โ algebraic multiplicity = geometric multiplicity
5.4. Complex Eigenvalues
6. Orthogonality
6.1. Orthogonal Complements
- dim Col A = dim Row A
- Row A = Col AT
- (Col A)โฅ^ = Nul AT
- (Nul A)โฅ^ = Col AT
6.2. Orthogonal Projection
- ATAv = ATx โ xW = Av 7. Summary
A = m x n Lives in Dimension
Span of columns of A Rm
Number of pivots
Codomain of T
Solution set of Ax = 0 Rn
Number of free variables
Domain of T
A = n x n has pivots in every column โ Col(A) = Rn โ Dim(Col(A)) = n = rank A โ Nul(A) = {0} โ Nullity A = 0 โ Columns of A are linearly independent โ Columns of A spans Rn โ Columns of A form a basis for Rn โ Ax = 0 has only the trivial solution โ Ax = b is consistent for all b in Rn โ Ax = b has a unique solution for each b in Rn โ T is one-to-one โ T is onto โ A is invertible (A-1^ exists) โ T is invertible โ det A โ 0 (if det A = 0, A-1^ is undefined)
8. Commonly made mistakes - If u, v, w are linearly independent, u, u - v, u - v + 2w are also linearly independent - If v 1 , v 2 , v 3 are linearly independent vectors in Rn, x1v1 + x2v2 + x3v3 = b has exactly one solution only if b is in span {v 1 , v 2 , v 3 } - If u, v are vectors in subspace W, 2u - 5v is also a vector in W
