Quadratic forms | Linear algebra

Null vector

In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: The null cone is also the union of the isotropic lines through the origin. (Wikipedia).

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Null Space: Is a Vector in a Null Space? Find a Basis for a Null Space

This video explains how to determine if a vector is in a null space and how to find a basis for a null space.

From playlist Column and Null Space

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Prove a Null Space is a Subspace

This video provides a basic proof that a null space is a subspace.

From playlist Column and Null Space

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Null space of a matrix using sympy

In this video I give a brief overview of the null space of a matrix and how to calculate (y hand) the column vectors that will make up the null space of the column. This becomes much easier in sympy, by simply using the nullspace function. If you want to learn more about the null space o

From playlist Modern linear algebra using Python instead of a textbook

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The Largest and Smallest Values for the Rank and Nullity of a Matrix (3 x 5)

This video explains how to determine the largest and smallest possible values for the rank and nullity of a 3 by 5 matrix.

From playlist Column and Null Space

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Null space of a matrix example

In today's lecture I work through an example to show you a well-known pitfall when it comes to the null space of a matrix. In the example I show you how to create the special cases and how to use them to represent the null space. There is also a quick look at the NullSpace function in Ma

From playlist Introducing linear algebra

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Null space of a matrix

In this video I start to discuss the idea of the null space of a matrix. In these situations, the right-hand side of all the equations in the linear system is equal to zero. There is the trivial solution, where all the elements of the solution is zero. We are more interested in the spec

From playlist Introducing linear algebra

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Null Space and Column Space of a Matrix

Given a matrix A(ie a linear transformation) there are several important related subspaces. In this video we investigate the Nullspace of A and the column space of A. The null space is the vectors that are "killed" by the transformation - ie sent to zero. The column space will be the image

From playlist Older Linear Algebra Videos

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Find a Spanning Vectors (Basis) of the Null Space of a Matrix (Two Vectors)

This video explains how to find a basis or spanning vectors for the null space of a matrix.

From playlist Column and Null Space

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Linear Algebra - Lecture 28 - Null Space of a Matrix

In this video lecture, I define the null space of a matrix, and work through several examples.

From playlist Linear Algebra Lectures

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Linear Algebra 15o: The Null Space of a Linear Transformation

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

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Linear Algebra 15p: The Inverse of a Linear Transformation

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 3 Linear Algebra: Linear Transformations

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Review of Ax=b for Differential Equations

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Partial Differential Equations

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9. Independence, Basis, and Dimension

MIT 18.06 Linear Algebra, Spring 2005 Instructor: Gilbert Strang View the complete course: http://ocw.mit.edu/18-06S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 9. Independence, Basis, and Dimension License: Creative Commons BY-NC-SA More information at h

From playlist MIT 18.06 Linear Algebra, Spring 2005

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Visualizations of left nullspace and rowspace | Linear Algebra | Khan Academy

Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/linear-algebra/matrix-transformations/matrix-transpose/v/lin-alg-visualizations-of-left-nullspace-and-rowspace Relationship between left nullspace, rowspace, colum

From playlist Matrix transformations | Linear Algebra | Khan Academy

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Linear Algebra 7f2: 21 Easy Null Space Exercises

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 1 Linear Algebra: An In-Depth Introduction with a Focus on Applications

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The Largest and Smallest Values for the Rank and Nullity of a Matrix (5 x 3)

This video explains how to determine the largest and smallest possible values for the rank and nullity of a 5 by 3 matrix.

From playlist Column and Null Space

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Null space of a matrix | Lecture 21 | Matrix Algebra for Engineers

Definition of the null space of a matrix and how to compute it. Join me on Coursera: https://www.coursera.org/learn/matrix-algebra-engineers Lecture notes at http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/user/jchasnov?sub

From playlist Matrix Algebra for Engineers

Related pages

Newman–Penrose formalism | Basis (linear algebra) | Bilinear form | Isotropic quadratic form | Definite quadratic form | Quadratic form | Pseudo-Euclidean space | Vector space | Mathematics | Division algebra | Biquaternion | Real number | Verma module | Lie algebra | Isotropic line | Composition algebra | Minkowski space