M1: Linear Algebra I (2021-22)
Main content blocks
- Lecturer: Profile: Richard Earl
(i) use the definitions of a vector space, a subspace, linear dependence and independence, spanning sets and bases, both within the familiar setting of \(\mathbb{R}^2\) and \(\mathbb{R}^3\) and also for abstract vector spaces, and prove results using these definitions;
(ii) use matrices to solve systems of linear equations and to determine the number of solutions of such a system;
(iii) solve a range of problems relating to linear maps between vector spaces, thinking of linear maps abstractly or representing them using matrices as appropriate.
Matrices and the beginnings of matrix algebra.
Use of matrices to describe systems of linear equations.
Elementary Row Operations (EROs) on matrices.
Reduction of matrices to echelon form.
Application to the solution of systems of linear equations.
Inverse of a square matrix.
Reduced row echelon (RRE) form and the use of EROs to compute inverses; computational efficiency of the method.
Transpose of a matrix;
orthogonal matrices.
Vector spaces: definition of a vector space over a field (such as
\(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{C}\)).
Subspaces.
Many explicit examples of vector spaces and subspaces.
Span of a set of vectors.
Examples such as row space and column space of a matrix.
Linear dependence and independence.
Bases of vector spaces; examples.
The Steinitz Exchange Lemma; dimension.
Application to matrices: row space and column space, row rank and column rank.
Coordinates associated with a basis of a vector space.
Use of EROs to find bases of subspaces.
Sums and intersections of subspaces; the dimension formula.
Direct sums of subspaces.
Linear transformations: definition and examples (including projections associated with direct-sum decompositions).
Some algebra of linear transformations; inverses.
Kernel and image, Rank-Nullity Theorem.
Applications including algebraic characterisation of projections (as idempotent linear transformations).
Matrix of a linear transformation with respect to bases.
Change of Bases Theorem.
Applications including proof that row rank and column rank of a matrix are equal.
Bilinear forms; real inner product spaces; examples.
Mention of complex inner product spaces.
Cauchy--Schwarz inequality.
Distance and angle.
The importance of orthogonal matrices.
Section outline
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The 14 lectures will cover the material as broken down below:
1-3: Linear Systems, Matrix Algebra
3-4: Inverses and Transposes
4-5: Vector Spaces and Subspaces
6: Bases
7: Dimension
8: Dimension and Subspaces
9-10: Linear Maps. Rank-Nullity Theorem
11-12: Matrices representing Linear Maps
13-14: Inner Product Spaces
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This sheet corresponds to material covered in lectures 1 and 2, namely
Systems of linear equations
Matrix algebra
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These problems correspond to material covered in lectures 3 and 4, in particular
RRE form
Transposes and Inverses
Invertibility of EROs
Process with EROs for determining invertibility
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These problems correspond to material covered in lectures 5 and 6, in particular
Vector spaces and subspaces
Linearly independent sets
Spanning sets
Bases
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These problems correspond to material covered in lectures 7 and 8, in particular
Bases
Dimension
The dimension formula
Direct Sums
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These problems correspond to material covered in lectures 9 and 10, in particular
Linear Transformations
Kernels
Images
Rank-Nullity Theorem
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These problems correspond to material covered in lectures 11 and 12, in particular
Matrices representing Linear Transformations
Change of Basis
Row and Column Rank
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These problems correspond to material covered in lectures 13 and 14, in particular
Bilinear Forms
Inner Product Spaces
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A historical dissertation on the ancient Chinese methods for solving linear systems of equations.