Rank of Matrix (Row Echelon Form)
Calc Find Rank by reducing to Row Echelon Form.
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Method:
1. Use elementary row operations (R2 -> R2 - kR1) to make zeros below pivot.
2. Convert to Upper Triangular Matrix.
3. Rank = Number of Non-Zero Rows.
Specific Matrices Asked:
[1 2 -3 1 -6; 1 1 2 -1 7...]
[3 2 -3 1 -6; -1 3 2 -2 5...]
[0 1 -3 -1; 1 0 1 1...]
System of Linear Equations
High Yield Consistency: Find values of α, β (or a, b) for Unique/Infinite/No Solution.
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Conditions:
• Unique: Rank(A) = Rank(A|B) = n (variables).
• Infinite: Rank(A) = Rank(A|B) < n.
• No Solution: Rank(A) ≠ Rank(A|B).
Calc Inverse of Matrix using Gauss-Jordan Method.
Pg 2
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Method:
1. Augment matrix A with Identity matrix I: [A | I].
2. Apply row operations to convert A into I: [I | A⁻¹].
3. The right side is the Inverse.
Subspaces & Vector Space Axioms
Concept Check if set W is a Subspace/Vector Space.
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Subspace Test:
1. Is Zero vector in W?
2. Is W closed under addition? (u+v in W)
3. Is W closed under scalar multiplication? (ku in W)
Linear Independence & Basis
High Yield Basis Extension & Reduction.
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Extension: Add standard basis vectors to the set, check independence, form matrix, reduce to echelon, pick pivot columns.
Reduction: Form matrix with vectors as rows, reduce to echelon, non-zero rows form basis.
Transition Matrix & Coordinates
Calc Transition Matrix (P) & Coordinate Vectors [w].
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Transition Matrix P (B' to B):
[ [v1]_B [v2]_B ... [vn]_B ]
Coordinate Change: [x]_B = P * [x]_B'
Standard Matrices & Geometry
Calc Find Standard Matrix (Rotation, Dilation, Reflection).
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Rotation (2D): [[cosθ -sinθ], [sinθ cosθ]]
Dilation (k): [[k 0], [0 k]]
Reflection (x-axis): [[1 0], [0 -1]]
Kernel, Range & Inverse
Concept Kernel, Range & Dimension Theorem.
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Dimension Theorem (Rank-Nullity):
Dim(V) = Rank(T) + Nullity(T)
Kernel: Solve T(x)=0.
Range: Col space of Standard Matrix.
Calc Inverse of Linear Transformation.
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Steps:
1. Find standard matrix A.
2. Find A⁻¹.
3. Define T⁻¹ using A⁻¹.
Eigenvalues & Cayley-Hamilton
High Yield Algebraic & Geometric Multiplicity.
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Algebraic (AM): Number of times λ repeats as root.
Geometric (GM): Dim(Eigenspace) = n - Rank(A-λI).
Matrix is diagonalizable iff AM = GM for all λ.
Calc Verify Cayley-Hamilton Theorem. Find A⁻¹ or Aⁿ.
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Theorem: Every square matrix satisfies its own characteristic equation.
If λ³ - 6λ² + 11λ - 6 = 0, then A³ - 6A² + 11A - 6I = 0.
Multiply by A⁻¹ to find Inverse.
Diagonalization
Long Diagonalize Matrix A (Find P).
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P = [v1 v2 v3] (Eigenvectors)
D = P⁻¹AP (Diagonal matrix with Eigenvalues)