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Linear combination and vector equations Rn
- Intro Lesson: a9:28
- Intro Lesson: b5:58
- Intro Lesson: c4:22
- Intro Lesson: d26:20
- Lesson: 1a2:50
- Lesson: 1b3:01
- Lesson: 1c3:25
- Lesson: 26:38
- Lesson: 35:05
- Lesson: 4a13:25
- Lesson: 4b18:30
- Lesson: 4c5:22
- Lesson: 513:07
Linear combination and vector equations Rn
Basic Concepts: Introduction to vectors, Scalar multiplication, Adding and subtracting vectors in component form
Lessons
A matrix with one column is called a column vector. They can be added or subtracted with other column vectors as long as they have the same amount of rows.
Parallelogram Rule for Addition: if you have two vectors u and v, then u+v would be the fourth vertex of a parallelogram whose other vertices are u,(0,0),and v
Here are the following algebraic properties of Rn
1. u+v=v+u
2. (u+v)+w=u+(v+w)
3. u+0=0+u=u
4. u+(−u)=−u+u=0
5. c(u+v)=cu+cv
6. (c+d)u=cu+du
7. c(du)=(cd)(u)
8. 1u=u
Given vectors v1,⋯,vp in Rn with scalars c1,⋯,cp, the vector x is defined by
x=v1c1+⋯+vpcp
Where x is a linear combination of v1,⋯,vp.
The linear combinations of v1,⋯,vp is the same as saying Span{v1,⋯,vp}.
Parallelogram Rule for Addition: if you have two vectors u and v, then u+v would be the fourth vertex of a parallelogram whose other vertices are u,(0,0),and v
Here are the following algebraic properties of Rn
1. u+v=v+u
2. (u+v)+w=u+(v+w)
3. u+0=0+u=u
4. u+(−u)=−u+u=0
5. c(u+v)=cu+cv
6. (c+d)u=cu+du
7. c(du)=(cd)(u)
8. 1u=u
Given vectors v1,⋯,vp in Rn with scalars c1,⋯,cp, the vector x is defined by
Where x is a linear combination of v1,⋯,vp.
The linear combinations of v1,⋯,vp is the same as saying Span{v1,⋯,vp}.
- IntroductionVector Equations in Rn Overview:a)Vectors in R2
• Column vectors with 2 rows
• Adding, subtracting, and multiplying 2D vectors
• Graphing vectors in 2D
• Parallelogram Rule for Additionb)Vectors in R3
• Column vectors with 3 rows
• Adding, subtracting, and multiplying 3D vectors
• Graphing vectors in 3D
c)Vectors in Rn
• Column vector with n rows
• Algebraic propertiesd)Linear Combinations and Spans
• Vectors and weights
• Vector equations
• Finding a linear combination with row reduction - 1.Calculating Vectors in Rn
Consider the two vectors, and
. Compute:
a)u+2vb)2u−vc)5u+0v - 2.Converting Systems Of Equations And Vector Equations
Write the given systems of equations as a vector equation.
2x1+x2−5x3=4 x1+3x2+2x3=1 −4x1−x2−8x3=−2 - 3.Write the given vector equation has a system of equations
- 4.Linear Combinations with Known terms
Determine if b is a linear combination of a1, a2 in part a. Determine if b is a linear combination of a1, a2, and a3 in part b and c.a)b)c) - 5.Linear Combinations with Unknown terms
For what value(s) of k is b in the plane spanned by a1 and a2 if:
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1.
Linear Equations with Matrices
1.1
Notation of matrices
1.2
Solving systems of linear equations by graphing
1.3
Representing linear system as a matrix
1.4
The three types of matrix row operations
1.5
Solving a linear system with matrices using Gaussian elimination
1.6
Row reduction and echelon forms
1.7
Linear combination and vector equations in Rn
1.8
Matrix equation Ax=b
1.9
Solution sets of linear systems
1.10
Application of linear systems