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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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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…
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