filmov
tv
Advanced Linear Algebra, Lecture 3.3: Alternating multilinear forms

Показать описание
Advanced Linear Algebra, Lecture 3.3: Alternating multilinear forms
A multilinear form is alternating if it is zero whenever two distinct inputs are identical. We show how alternating k-linear forms are skew-symmetric, and the converse holds as long as we are over a field where 1+1≠ 0. After that, we show how if the input vectors to an alternating k-linear form are linearly dependent, the output will be zero. The converse fails -- there are k-linear forms that evaluate to zero on linearly independent sets. However, the converse holds in one important case: when k=n=dim(X), which is a property that we know to hold for determinants. The proof of this actually tells us more -- that any two alternating n-linear forms are scalar multiples of each other. The determinant will end up being the unique alternating n-linear form that is "normalized" to be 1 on the standard unit basis vectors, and this is the topic of the following lecture.
A multilinear form is alternating if it is zero whenever two distinct inputs are identical. We show how alternating k-linear forms are skew-symmetric, and the converse holds as long as we are over a field where 1+1≠ 0. After that, we show how if the input vectors to an alternating k-linear form are linearly dependent, the output will be zero. The converse fails -- there are k-linear forms that evaluate to zero on linearly independent sets. However, the converse holds in one important case: when k=n=dim(X), which is a property that we know to hold for determinants. The proof of this actually tells us more -- that any two alternating n-linear forms are scalar multiples of each other. The determinant will end up being the unique alternating n-linear form that is "normalized" to be 1 on the standard unit basis vectors, and this is the topic of the following lecture.
Advanced Linear Algebra, Lecture 3.3: Alternating multilinear forms
Advanced Linear Algebra 3: Bases
Advanced Linear Algebra - Lecture 3: Linear Combinations and Spans
Advanced Linear Algebra, Lecture 2.3: Algebra of linear mappings
Gilbert Strang: Linear Algebra vs Calculus
Grant Sanderson (3Blue1Brown): Best Way to Learn Math | AI Podcast Clips
Advanced Linear Algebra Lecturette 3: Inner Product Spaces
Advanced Linear Algebra 4: Dimension of a Vector Space
Advanced Linear Algebra, Lecture 3.2: Symmetric and skew-symmetric multilinear forms
Linear Algebra and it's Applications by Gilbert Strang #shorts
The Hardest Math Test
Legendary Book for Learning Abstract Algebra
Advanced Linear Algebra, Lecture 3.6: Minors and cofactors
How to Solve Linear Equations With Variables on Both Sides : Linear Algebra Education
01.2.3 The 2 norm
The Big Picture of Linear Algebra
Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra
Advanced Linear Algebra, Lecture 3.5: The determinant and trace of a matrix
Basic Algebra Tips
Super Hard Algebra Book
Advanced Linear Algebra 16: Adjoint of Linear Transformation
A Book of Abstract Algebra
Advanced Linear Algebra - Lecture 23: The Adjoint of a Linear Transformation
Exploring Abstract Algebra
Комментарии