CRS Algebra 1

CMI, Aug-Nov 2024, Aditya Karnataki
Treil (2014), Artin (2011), Hoffman & Kunze (2014), Axler (2015), Curtis (1999)

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• Vector spaces
• Matrices
• Linear Combinations
• Bases
• Linear Transformations
• Trace
• Invertible Transformations
• Isomorphism
• Subspaces
• Solving linear systems, Pivots
• How to find matrices with a given kernel

Lecture Notes

• LEC ALG1 8 ✅
  • Every fdvsp has a basis.
  • Any two bases of an fdvsp have same cardinality, i.e, the cardinality of a basis is an invariant of an fdvsp.
• LEC ALG1 9 ✅
  • Working with bases, finding a basis for the null space of a matrix
• LEC ALG1 10 ✅
  • Finding a basis for the column space of a matrix, equivalence of column rank and row rank, rank nullity theorem for matrices
• LEC ALG1 11 ✅
  • Linear maps, rank nullity theorem for linear maps over abstract vector spaces
• LEC ALG1 12 ✅
  • Linear maps in ${\mathbb{R}}^n$ can be represented as matrices, matrices of linear maps between abstract vector spaces, change of basis
• LEC ALG1 13 ✅
  • Homomorphisms, more change of basis, composition of linear maps in terms of matrices, choosing a good basis for a linear map
• LEC ALG1 14 ✅
  • Sums of subspaces, direct sums, dimension of a sum, determinants
• LEC ALG1 15 ✅
  • An algorithm to compute the determinant, multilinearity and alternate characterization of the determinant, cofactor expansions
• LEC ALG1 16 ✅
  • Alternate formula for determinant and proof of its uniqueness, properties of determinant, Invariant subspaces
• LEC ALG1 17 ✅
  • Eigenvectors, eigenvalues, eigenspaces
• LEC ALG1 18 ✅
  • Finding Eigenstuff of matrices and abstract operators, characteristic polynomial
• LEC ALG1 19 ✅
  • Diagonalization
• LEC ALG1 20 ✅
  • Dual spaces, canonical isomorphisms, introduction to inner product spaces
• LEC ALG1 21 ✅
  • Inner product spaces, normed spaces, orthogonal vectors, Gram-Schmidt orthogonalization process
• LEC ALG1 22 ✅
  • Gram-Schmidt example, orthogonal decomposition theorem
• LEC ALG1 23

Excalidraw

These notes are either in excalidraw (which cannot be rendered by Quartz) or unformatted.

• ALG1_L1 Intro, Vector spaces, Fields.
• ALG1_L2 Matrices, Variables vs equations table
• ALG1_L3 Row operations, REF, RREF, pivots, free variables, conditions for $A\mathbf{x}=\mathbf{b}$ being inconsistent and consistent
• ALG1_L4
  • Given $\mathbf{v}$ s.t $A\mathbf{v}=\mathbf{b}$, any other $\mathbf{w}$ s.t $A\mathbf{w}=\mathbf{b}$ can be expressed as $\mathbf{v}+\mathbf{y}$, $\mathbf{y}\in \text{Ker }A$.
  • requirements for injectivity and surjectivity in terms of number of pivots
  • matrix representation of row ops
  • matrix multiplication as a function
  • subspaces
• ALG1_L5
  • injectivity and surjectivity requirements in terms of number of rows and columns
  • Invertible matrices
  • If $A$ is invertible, its inverse must be unique
  • If $A$ is invertible, it must be square
  • Vector spaces, subspaces of ${\mathbb{R}}^2$.
• LEC ALG1 6
  • linear combinations, span. function surjectivity relation with spanning property of column vectors, function injectivity relation with unique linear combination of column vectors.
  • linear independence
• LEC ALG1 7
  • Any set $S\subset {\mathbb{R}}^n$ whose span equals ${\mathbb{R}}^n$ must be of size $\ge n$
  • Any linearly independent set in ${\mathbb{R}}^n$ must be of size $\le n$.
  • basis
  • defined dimension as cardinality of basis
  • posed question: does every vector space have a basis?
  • fdvsp
• TUT ALG1 1
• TUT ALG1 2

Homework

• ALG1_HW1
• ALG1_HW2
• ALG1_HW3
• ALG1_HW4
• ALG1_HW5

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References

Artin, M. (2011). Algebra (2. ed). Pearson Education, Prentice Hall.

Axler, S. (2015). Linear Algebra Done Right. Springer International Publishing. https://doi.org/10.1007/978-3-319-11080-6

Curtis, C. W. (1999). Linear Algebra: An Introductory Approach (Corr. 7. pr). Springer.

Hoffman, K., & Kunze, R. A. (2014). Linear Algebra (Second edition). PHI Learning Private Limited.

Treil, S. (2014). Linear Algebra Done Wrong.