الجبر الخطي | Linear algebra — مقرر جامعي مجاني
مِسبار · مقرر جامعي مجاني
Linear algebra
تعلم الجبر الخطي (Linear algebra) مجانًا عبر 3 وحدات و136 فيديو تعليميًا، بإجمالي نحو 33 ساعة، ضمن مكتبة مِسبار الجامعية.
Learn linear algebra—vectors, matrices, transformations, and more.
- 3 وحدات.
- 136 فيديو.
- نحو 33 ساعة مشاهدة.
- المستوى: college.
ابدأ مشاهدة المقرر داخل مكتبة مِسبار
التخصصات المرتبطة بالمقرر
- الهندسة الكهربائية (Electrical Engineering)
- هندسة الحاسوب (Computer Engineering)
- الهندسة الميكانيكية (Mechanical Engineering)
- الهندسة المدنية (Civil Engineering)
- الهندسة الكيميائية (Chemical Engineering)
- الهندسة الصناعية (Industrial Engineering)
- هندسة الطيران والفضاء (Aerospace Engineering)
- علوم الحاسوب (Computer Science)
- الذكاء الاصطناعي (Artificial Intelligence)
- علم البيانات (Data Science)
- الروبوتات (Robotics)
- الأمن السيبراني (Cybersecurity)
- الرياضيات (Mathematics)
- الإحصاء (Statistics)
- الفيزياء (Physics)
- تعليم الرياضيات (Mathematics Education)
وحدات ودروس المقرر
1. Vectors and spaces
41 فيديو تعليمي.
موضوعات ودروس الوحدة
- Vector intro for linear algebra
- Real coordinate spaces
- Adding vectors algebraically & graphically
- Multiplying a vector by a scalar
- Vector examples
- Unit vectors intro
- Parametric representations of lines
- Linear combinations and span
- Introduction to linear independence
- More on linear independence
- Span and linear independence example
- Linear subspaces
- Basis of a subspace
- Vector dot product and vector length
- Proving vector dot product properties
- Proof of the Cauchy-Schwarz inequality
- Vector triangle inequality
- Defining the angle between vectors
- Defining a plane in R3 with a point and normal vector
- Cross product introduction
- Proof: Relationship between cross product and sin of angle
- Dot and cross product comparison/intuition
- Vector triple product expansion (very optional)
- Normal vector from plane equation
- Point distance to plane
- Distance between planes
- Solving a system of 3 equations and 4 variables using matrix row-echelon form
- Solving linear systems with matrices
- Using matrix row-echelon form in order to show a linear system has no solutions
- Matrix vector products
- Introduction to the null space of a matrix
- Null space 2: Calculating the null space of a matrix
- Null space 3: Relation to linear independence
- Column space of a matrix
- Null space and column space basis
- Visualizing a column space as a plane in R3
- Proof: Any subspace basis has same number of elements
- Dimension of the null space or nullity
- Dimension of the column space or rank
- Showing relation between basis cols and pivot cols
- Showing that the candidate basis does span C(A)
2. Matrix transformations
56 فيديو تعليمي.
موضوعات ودروس الوحدة
- A more formal understanding of functions
- Vector transformations
- Linear transformations
- Matrix vector products as linear transformations
- Linear transformations as matrix vector products
- Image of a subset under a transformation
- im(T): Image of a transformation
- Preimage of a set
- Preimage and kernel example
- Sums and scalar multiples of linear transformations
- More on matrix addition and scalar multiplication
- Linear transformation examples: Scaling and reflections
- Linear transformation examples: Rotations in R2
- Rotation in R3 around the x-axis
- Unit vectors
- Introduction to projections
- Expressing a projection on to a line as a matrix vector prod
- Compositions of linear transformations 1
- Compositions of linear transformations 2
- Matrix product examples
- Matrix product associativity
- Distributive property of matrix products
- Introduction to the inverse of a function
- Proof: Invertibility implies a unique solution to f(x)=y
- Surjective (onto) and injective (one-to-one) functions
- Relating invertibility to being onto and one-to-one
- Determining whether a transformation is onto
- Exploring the solution set of Ax = b
- Matrix condition for one-to-one transformation
- Simplifying conditions for invertibility
- Showing that inverses are linear
- Deriving a method for determining inverses
- Example of finding matrix inverse
- Formula for 2x2 inverse
- 3 x 3 determinant
- n x n determinant
- Determinants along other rows/cols
- Rule of Sarrus of determinants
- Determinant when row multiplied by scalar
- (correction) scalar multiplication of row
- Determinant when row is added
- Duplicate row determinant
- Determinant after row operations
- Upper triangular determinant
- Simpler 4x4 determinant
- Determinant and area of a parallelogram
- Determinant as scaling factor
- Transpose of a matrix
- Determinant of transpose
- Transpose of a matrix product
- Transposes of sums and inverses
- Transpose of a vector
- Rowspace and left nullspace
- Visualizations of left nullspace and rowspace
- rank(a) = rank(transpose of a)
- Showing that A-transpose x A is invertible
3. Alternate coordinate systems (bases)
39 فيديو تعليمي.
موضوعات ودروس الوحدة
- Orthogonal complements
- dim(v) + dim(orthogonal complement of v) = n
- Representing vectors in rn using subspace members
- Orthogonal complement of the orthogonal complement
- Orthogonal complement of the nullspace
- Unique rowspace solution to Ax = b
- Rowspace solution to Ax = b example
- Projections onto subspaces
- Visualizing a projection onto a plane
- A projection onto a subspace is a linear transformation
- Subspace projection matrix example
- Another example of a projection matrix
- Projection is closest vector in subspace
- Least squares approximation
- Least squares examples
- Another least squares example
- Coordinates with respect to a basis
- Change of basis matrix
- Invertible change of basis matrix
- Transformation matrix with respect to a basis
- Alternate basis transformation matrix example
- Alternate basis transformation matrix example part 2
- Changing coordinate systems to help find a transformation matrix
- Introduction to orthonormal bases
- Coordinates with respect to orthonormal bases
- Projections onto subspaces with orthonormal bases
- Finding projection onto subspace with orthonormal basis example
- Example using orthogonal change-of-basis matrix to find transformation matrix
- Orthogonal matrices preserve angles and lengths
- The Gram-Schmidt process
- Gram-Schmidt process example
- Gram-Schmidt example with 3 basis vectors
- Introduction to eigenvalues and eigenvectors
- Proof of formula for determining eigenvalues
- Example solving for the eigenvalues of a 2x2 matrix
- Finding eigenvectors and eigenspaces example
- Eigenvalues of a 3x3 matrix
- Eigenvectors and eigenspaces for a 3x3 matrix
- Showing that an eigenbasis makes for good coordinate systems
مقررات مرتبطة
- حساب المثلثات | Trigonometry
- ما قبل التفاضل والتكامل | Precalculus
- الإحصاء والاحتمالات | Statistics & probability
- التفاضل والتكامل AP (AB) | AP®︎ Calculus AB
- التفاضل والتكامل AP (BC) | AP®︎ Calculus BC
- الإحصاء AP | AP®︎ Statistics
المصدر والحقوق
المحتوى الأصلي المرخص: أكاديمية خان. تقوم مِسبار بالفهرسة والتنظيم حسب التخصص، وتعرض الفيديوهات المرخصة غير التجارية داخل مكتبة مجانية مستقلة.