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Week 12: Systems of Differential Equations

We have already worked with first-order linear differential equations. But what if there are more differential equations, and they are coupled, so they cannot be solved individually? Today we shall…

From  Jesper Kampmann Larsen on November 25th, 2020 0 plays 112  

Week 12: Systems of Differential Equations part 1

We have already worked with first-order linear differential equations. But what if there are more differential equations, and they are coupled, so they cannot be solved individually? Today we shall…

From  Jesper Kampmann Larsen on November 25th, 2020 0 plays 112  

Week 11, Short Day: Diagonalization by Orthogonal Substitution

We continue by investigating particular properties of symmetric matrices and their diagonalization. More precisely, what does it mean that the diagonalizing matrix is positive orthogonal. At the end…

From  Jesper Kampmann Larsen on November 20th, 2020 0 plays 112  

Week 11: Symmetric Matrices part 2

The starting point for talking about orthogonal vectors in Rn and orthogonal matrices in Rn×n is the scalar product, popularly known as the dot product. This gives us the opportunity to…

From  Jesper Kampmann Larsen on November 18th, 2020 0 plays 0  

Week 11: Symmetric Matrices part 1

The starting point for talking about orthogonal vectors in Rn and orthogonal matrices in Rn×n is the scalar product, popularly known as the dot product. This gives us the opportunity to…

From  Jesper Kampmann Larsen on November 18th, 2020 0 plays 0  

Week 10 Short Day: Diagonalization by Similarity Transform

If a square matrix is similar to a diagonal matrix, it is said to be diagonalizable by a similarity transformation. This is closely connected to the eigenvalue problem. Today we investigate the…

From  Jesper Kampmann Larsen on November 13th, 2020 0 plays 109  

Week 10 Long Day: Eigenvalues and Eigenvectors - part 2

When you investigate a linear map f:V→V of a vector space into itself a special question arises: Is there proper vectors (i.e. vectors different from the zero-vector), whose image vector is…

From  Jesper Kampmann Larsen on November 11th, 2020 0 plays 0  

Week 10 Long Day: Eigenvalues and Eigenvectors - part 1

When you investigate a linear map f:V→V of a vector space into itself a special question arises: Is there proper vectors (i.e. vectors different from the zero-vector), whose image vector is…

From  Jesper Kampmann Larsen on November 11th, 2020 0 plays 109  

Week 8 Short Day: Mapping Matrices and Change of Base

We continue to work with linear maps and their mapping matrices and we shall study typical examples on how you can determine their kernel an images. Important: When the basis is changed in the domain…

From  Jesper Kampmann Larsen on October 30th, 2020 0 plays 107  

Week 8, Long Day: Linear Maps (Part 1)

You know elementary functions y=f(x) that to every real number x attaches a real number y. Todays subject is linear maps y=f(x) that to a vector x attaches a vector y. We shall investigate how known…

From  Jesper Kampmann Larsen on October 28th, 2020 0 plays 0  

Week 8, Long Day: Linear Maps (Part 2)

You know elementary functions y=f(x) that to every real number x attaches a real number y. Todays subject is linear maps y=f(x) that to a vector x attaches a vector y. We shall investigate how known…

From  Jesper Kampmann Larsen on October 28th, 2020 0 plays 0  

Week 7, Long Day: Vector Spaces part 2

Today we shall work with general vector spaces. Vector spaces are sets of widely different mathematical objects, that have some decisive properties in common. You should likely experience that…

From  Jesper Kampmann Larsen on October 21st, 2020 4 plays 0  

Week 7, Long Day: Vector Spaces part 1

Today we shall work with general vector spaces. Vector spaces are sets of widely different mathematical objects, that have some decisive properties in common. You should likely experience that…

From  Jesper Kampmann Larsen on October 21st, 2020 9 plays 0  

Week 6, Short Day: Geometry in the Plane and in Space

Today we continue to manage concepts such as linear combinations, linear independence, basis and coordinates. The day will also cause a repetition of knowledge from highschool about vectors in…

From  Jesper Kampmann Larsen on October 9th, 2020 26 plays 76  

Week 6, Long Day: Determinants and Vector Geometry (Part 1)

To every square matrix a special number - called the determinant - is attached. How do you compute this number, and what does the number say about the matrix? Today we start with these questions and…

From  Jesper Kampmann Larsen on October 7th, 2020 25 plays 0  

Week 6, Long Day: Determinants and Vector Geometry (Part 2)

To every square matrix a special number - called the determinant - is attached. How do you compute this number, and what does the number say about the matrix? Today we start with these questions and…

From  Jesper Kampmann Larsen on October 7th, 2020 32 plays 74