Mathematics II
Semester II · BCA 154 · 3 credits
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chapters done
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The syllabus asks for
- Matrices and operations
- Determinants
- Inverse of a matrix
- The determinant as an area
- Eight uses of matrices in computing
Key points
- Matrix multiplication combines rows with columns, is associative but not commutative.
- The determinant measures scaling and invertibility; row operations compute it efficiently.
- A matrix is invertible exactly when its determinant is non-zero; inverses come from the adjoint or Gauss–Jordan.
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The syllabus asks for
- Systems and their solutions
- Gaussian elimination
- Rank and consistency
- Cramer's rule
- Applications and numerical issues
- Three pictures of a linear system
- Eight uses of linear systems
Key points
- Elimination reduces any system to echelon form and then solves by back-substitution.
- Rank decides the number of solutions; free variables parametrise infinite families.
- Cramer's rule and inverses give formulas but elimination is the practical method.
- Applications range from curve fitting to chemistry, circuits and regression.
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The syllabus asks for
- Vector spaces and subspaces
- Span, independence, basis and dimension
- The four subspaces and rank–nullity
- Linear transformations
- Inner products and orthogonality
- The four subspaces in one picture
- Eight uses of vector spaces and linear maps
Key points
- Vector spaces abstract the idea of vector; subspaces, span, independence, basis and dimension organise them.
- Rank–nullity ties the solutions of A X = 0 to the rank of A.
- Linear maps are matrices; their kernel and image measure what is lost and what is reached.
- Orthogonal bases and projections give coordinates, Gram–Schmidt and least squares.
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The syllabus asks for
- Eigenvalues and eigenvectors
- Diagonalisation
- Applications
- What an eigenvector looks like
- Eight uses of eigenvalues
Key points
- Eigenvectors are the directions a matrix only stretches; eigenvalues are the stretch factors.
- The characteristic polynomial finds them; trace and determinant give quick checks.
- Diagonalisation turns matrix powers into powers of numbers; symmetric matrices diagonalise orthogonally.
- Markov chains, PageRank, PCA, stability and vibration analysis are all eigenvalue problems.
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The syllabus asks for
- Vectors in three dimensions
- Lines in space
- Planes
- Spheres and applications
- A plane, its normal and the distance to a point
- Eight uses of vectors in space
Key points
- The dot product measures alignment and angles; the cross product measures area and gives normals.
- The triple product measures volume and tests coplanarity.
- Lines and planes have vector, parametric and Cartesian forms, with simple distance and angle formulas.
- The same operations drive lighting, culling and reflection in graphics, and torque and work in physics.
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TU assessment: 60% final examination set by the university, 40% internal (assignments, lab work, attendance). Confirm the split in your college's notice.