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Raffi Hovasapian
Spanning Set for a Vector Space
Slide Duration:Table of Contents
I. Linear Equations and Matrices
Linear Systems
39m 3s
 Intro0:00
 Linear Systems1:20
 Introduction to Linear Systems1:21
 Examples10:35
 Example 110:36
 Example 213:44
 Example 316:12
 Example 423:48
 Example 528:23
 Example 632:32
 Number of Solutions35:08
 One Solution, No Solution, Infinitely Many Solutions35:09
 Method of Elimination36:57
 Method of Elimination36:58
Matrices
30m 34s
 Intro0:00
 Matrices0:47
 Definition and Example of Matrices0:48
 Square Matrix7:55
 Diagonal Matrix9:31
 Operations with Matrices10:35
 Matrix Addition10:36
 Scalar Multiplication15:01
 Transpose of a Matrix17:51
 Matrix Types23:17
 Regular: m x n Matrix of m Rows and n Column23:18
 Square: n x n Matrix With an Equal Number of Rows and Columns23:44
 Diagonal: A Square Matrix Where All Entries OFF the Main Diagonal are '0'24:07
 Matrix Operations24:37
 Matrix Operations24:38
 Example25:55
 Example25:56
Dot Product & Matrix Multiplication
41m 42s
 Intro0:00
 Dot Product1:04
 Example of Dot Product1:05
 Matrix Multiplication7:05
 Definition7:06
 Example 112:26
 Example 217:38
 Matrices and Linear Systems21:24
 Matrices and Linear Systems21:25
 Example 129:56
 Example 232:30
 Summary33:56
 Dot Product of Two Vectors and Matrix Multiplication33:57
 Summary, cont.35:06
 Matrix Representations of Linear Systems35:07
 Examples35:34
 Examples35:35
Properties of Matrix Operation
43m 17s
 Intro0:00
 Properties of Addition1:11
 Properties of Addition: A1:12
 Properties of Addition: B2:30
 Properties of Addition: C2:57
 Properties of Addition: D4:20
 Properties of Addition5:22
 Properties of Addition5:23
 Properties of Multiplication6:47
 Properties of Multiplication: A7:46
 Properties of Multiplication: B8:13
 Properties of Multiplication: C9:18
 Example: Properties of Multiplication9:35
 Definitions and Properties (Multiplication)14:02
 Identity Matrix: n x n matrix14:03
 Let A Be a Matrix of m x n15:23
 Definitions and Properties (Multiplication)18:36
 Definitions and Properties (Multiplication)18:37
 Properties of Scalar Multiplication22:54
 Properties of Scalar Multiplication: A23:39
 Properties of Scalar Multiplication: B24:04
 Properties of Scalar Multiplication: C24:29
 Properties of Scalar Multiplication: D24:48
 Properties of the Transpose25:30
 Properties of the Transpose25:31
 Properties of the Transpose30:28
 Example30:29
 Properties of Matrix Addition33:25
 Let A, B, C, and D Be m x n Matrices33:26
 There is a Unique m x n Matrix, 0, Such That…33:48
 Unique Matrix D34:17
 Properties of Matrix Multiplication34:58
 Let A, B, and C Be Matrices of the Appropriate Size34:59
 Let A Be Square Matrix (n x n)35:44
 Properties of Scalar Multiplication36:35
 Let r and s Be Real Numbers, and A and B Matrices36:36
 Properties of the Transpose37:10
 Let r Be a Scalar, and A and B Matrices37:12
 Example37:58
 Example37:59
Solutions of Linear Systems, Part 1
38m 14s
 Intro0:00
 Reduced Row Echelon Form0:29
 An m x n Matrix is in Reduced Row Echelon Form If:0:30
 Reduced Row Echelon Form2:58
 Example: Reduced Row Echelon Form2:59
 Theorem8:30
 Every m x n Matrix is RowEquivalent to a UNIQUE Matrix in Reduced Row Echelon Form8:31
 Systematic and Careful Example10:02
 Step 110:54
 Step 211:33
 Step 312:50
 Step 414:02
 Step 515:31
 Step 617:28
 Example30:39
 Find the Reduced Row Echelon Form of a Given m x n Matrix30:40
Solutions of Linear Systems, Part II
28m 54s
 Intro0:00
 Solutions of Linear Systems0:11
 Solutions of Linear Systems0:13
 Example I3:25
 Solve the Linear System 13:26
 Solve the Linear System 214:31
 Example II17:41
 Solve the Linear System 317:42
 Solve the Linear System 420:17
 Homogeneous Systems21:54
 Homogeneous Systems Overview21:55
 Theorem and Example24:01
Inverse of a Matrix
40m 10s
 Intro0:00
 Finding the Inverse of a Matrix0:41
 Finding the Inverse of a Matrix0:42
 Properties of NonSingular Matrices6:38
 Practical Procedure9:15
 Step19:16
 Step 210:10
 Step 310:46
 Example: Finding Inverse12:50
 Linear Systems and Inverses17:01
 Linear Systems and Inverses17:02
 Theorem and Example21:15
 Theorem26:32
 Theorem26:33
 List of NonSingular Equivalences28:37
 Example: Does the Following System Have a Nontrivial Solution?30:13
 Example: Inverse of a Matrix36:16
II. Determinants
Determinants
21m 25s
 Intro0:00
 Determinants0:37
 Introduction to Determinants0:38
 Example6:12
 Properties9:00
 Properties 159:01
 Example10:14
 Properties, cont.12:28
 Properties 6 & 712:29
 Example14:14
 Properties, cont.18:34
 Properties 8 & 918:35
 Example19:21
Cofactor Expansions
59m 31s
 Intro0:00
 Cofactor Expansions and Their Application0:42
 Cofactor Expansions and Their Application0:43
 Example 13:52
 Example 27:08
 Evaluation of Determinants by Cofactor9:38
 Theorem9:40
 Example 111:41
 Inverse of a Matrix by Cofactor22:42
 Inverse of a Matrix by Cofactor and Example22:43
 More Example36:22
 List of NonSingular Equivalences43:07
 List of NonSingular Equivalences43:08
 Example44:38
 Cramer's Rule52:22
 Introduction to Cramer's Rule and Example52:23
III. Vectors in Rn
Vectors in the Plane
46m 54s
 Intro0:00
 Vectors in the Plane0:38
 Vectors in the Plane0:39
 Example 18:25
 Example 215:23
 Vector Addition and Scalar Multiplication19:33
 Vector Addition19:34
 Scalar Multiplication24:08
 Example26:25
 The Angle Between Two Vectors29:33
 The Angle Between Two Vectors29:34
 Example33:54
 Properties of the Dot Product and Unit Vectors38:17
 Properties of the Dot Product and Unit Vectors38:18
 Defining Unit Vectors40:01
 2 Very Important Unit Vectors41:56
nVector
52m 44s
 Intro0:00
 nVectors0:58
 4Vector0:59
 7Vector1:50
 Vector Addition2:43
 Scalar Multiplication3:37
 Theorem: Part 14:24
 Theorem: Part 211:38
 Right and Left Handed Coordinate System14:19
 Projection of a Point Onto a Coordinate Line/Plane17:20
 Example21:27
 CauchySchwarz Inequality24:56
 Triangle Inequality36:29
 Unit Vector40:34
 Vectors and Dot Products44:23
 Orthogonal Vectors44:24
 CauchySchwarz Inequality45:04
 Triangle Inequality45:21
 Example 145:40
 Example 248:16
Linear Transformation
48m 53s
 Intro0:00
 Introduction to Linear Transformations0:44
 Introduction to Linear Transformations0:45
 Example 19:01
 Example 211:33
 Definition of Linear Mapping14:13
 Example 322:31
 Example 426:07
 Example 530:36
 Examples36:12
 Projection Mapping36:13
 Images, Range, and Linear Mapping38:33
 Example of Linear Transformation42:02
Linear Transformations, Part II
34m 8s
 Intro0:00
 Linear Transformations1:29
 Linear Transformations1:30
 Theorem 17:15
 Theorem 29:20
 Example 1: Find L (3, 4, 2)11:17
 Example 2: Is It Linear?17:11
 Theorem 325:57
 Example 3: Finding the Standard Matrix29:09
Lines and Planes
37m 54s
 Intro0:00
 Lines and Plane0:36
 Example 10:37
 Example 27:07
 Lines in IR39:53
 Parametric Equations14:58
 Example 317:26
 Example 420:11
 Planes in IR325:19
 Example 531:12
 Example 634:18
IV. Real Vector Spaces
Vector Spaces
42m 19s
 Intro0:00
 Vector Spaces3:43
 Definition of Vector Spaces3:44
 Vector Spaces 15:19
 Vector Spaces 29:34
 Real Vector Space and Complex Vector Space14:01
 Example 115:59
 Example 218:42
 Examples26:22
 More Examples26:23
 Properties of Vector Spaces32:53
 Properties of Vector Spaces Overview32:54
 Property A34:31
 Property B36:09
 Property C36:38
 Property D37:54
 Property F39:00
Subspaces
43m 37s
 Intro0:00
 Subspaces0:47
 Defining Subspaces0:48
 Example 13:08
 Example 23:49
 Theorem7:26
 Example 39:11
 Example 412:30
 Example 516:05
 Linear Combinations23:27
 Definition 123:28
 Example 125:24
 Definition 229:49
 Example 231:34
 Theorem32:42
 Example 334:00
Spanning Set for a Vector Space
33m 15s
 Intro0:00
 A Spanning Set for a Vector Space1:10
 A Spanning Set for a Vector Space1:11
 Procedure to Check if a Set of Vectors Spans a Vector Space3:38
 Example 16:50
 Example 214:28
 Example 321:06
 Example 422:15
Linear Independence
17m 20s
 Intro0:00
 Linear Independence0:32
 Definition0:39
 Meaning3:00
 Procedure for Determining if a Given List of Vectors is Linear Independence or Linear Dependence5:00
 Example 17:21
 Example 210:20
Basis & Dimension
31m 20s
 Intro0:00
 Basis and Dimension0:23
 Definition0:24
 Example 13:30
 Example 2: Part A4:00
 Example 2: Part B6:53
 Theorem 19:40
 Theorem 211:32
 Procedure for Finding a Subset of S that is a Basis for Span S14:20
 Example 316:38
 Theorem 321:08
 Example 425:27
Homogeneous Systems
24m 45s
 Intro0:00
 Homogeneous Systems0:51
 Homogeneous Systems0:52
 Procedure for Finding a Basis for the Null Space of Ax = 02:56
 Example 17:39
 Example 218:03
 Relationship Between Homogeneous and NonHomogeneous Systems19:47
Rank of a Matrix, Part I
35m 3s
 Intro0:00
 Rank of a Matrix1:47
 Definition1:48
 Theorem 18:14
 Example 19:38
 Defining Row and Column Rank16:53
 If We Want a Basis for Span S Consisting of Vectors From S22:00
 If We want a Basis for Span S Consisting of Vectors Not Necessarily in S24:07
 Example 2: Part A26:44
 Example 2: Part B32:10
Rank of a Matrix, Part II
29m 26s
 Intro0:00
 Rank of a Matrix0:17
 Example 1: Part A0:18
 Example 1: Part B5:58
 Rank of a Matrix Review: Rows, Columns, and Row Rank8:22
 Procedure for Computing the Rank of a Matrix14:36
 Theorem 1: Rank + Nullity = n16:19
 Example 217:48
 Rank & Singularity20:09
 Example 321:08
 Theorem 223:25
 List of NonSingular Equivalences24:24
 List of NonSingular Equivalences24:25
Coordinates of a Vector
27m 3s
 Intro0:00
 Coordinates of a Vector1:07
 Coordinates of a Vector1:08
 Example 18:35
 Example 215:28
 Example 3: Part A19:15
 Example 3: Part B22:26
Change of Basis & Transition Matrices
33m 47s
 Intro0:00
 Change of Basis & Transition Matrices0:56
 Change of Basis & Transition Matrices0:57
 Example 110:44
 Example 220:44
 Theorem23:37
 Example 3: Part A26:21
 Example 3: Part B32:05
Orthonormal Bases in nSpace
32m 53s
 Intro0:00
 Orthonormal Bases in nSpace1:02
 Orthonormal Bases in nSpace: Definition1:03
 Example 14:31
 Theorem 16:55
 Theorem 28:00
 Theorem 39:04
 Example 210:07
 Theorem 213:54
 Procedure for Constructing an O/N Basis16:11
 Example 321:42
Orthogonal Complements, Part I
21m 27s
 Intro0:00
 Orthogonal Complements0:19
 Definition0:20
 Theorem 15:36
 Example 16:58
 Theorem 213:26
 Theorem 315:06
 Example 218:20
Orthogonal Complements, Part II
33m 49s
 Intro0:00
 Relations Among the Four Fundamental Vector Spaces Associated with a Matrix A2:16
 Four Spaces Associated With A (If A is m x n)2:17
 Theorem4:49
 Example 17:17
 Null Space and Column Space10:48
 Projections and Applications16:50
 Projections and Applications16:51
 Projection Illustration21:00
 Example 123:51
 Projection Illustration Review30:15
V. Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
38m 11s
 Intro0:00
 Eigenvalues and Eigenvectors0:38
 Eigenvalues and Eigenvectors0:39
 Definition 13:30
 Example 17:20
 Example 210:19
 Definition 221:15
 Example 323:41
 Theorem 126:32
 Theorem 227:56
 Example 429:14
 Review34:32
Similar Matrices & Diagonalization
29m 55s
 Intro0:00
 Similar Matrices and Diagonalization0:25
 Definition 10:26
 Example 12:00
 Properties3:38
 Definition 24:57
 Theorem 16:12
 Example 39:37
 Theorem 212:40
 Example 419:12
 Example 520:55
 Procedure for Diagonalizing Matrix A: Step 124:21
 Procedure for Diagonalizing Matrix A: Step 225:04
 Procedure for Diagonalizing Matrix A: Step 325:38
 Procedure for Diagonalizing Matrix A: Step 427:02
Diagonalization of Symmetric Matrices
30m 14s
 Intro0:00
 Diagonalization of Symmetric Matrices1:15
 Diagonalization of Symmetric Matrices1:16
 Theorem 12:24
 Theorem 23:27
 Example 14:47
 Definition 16:44
 Example 28:15
 Theorem 310:28
 Theorem 412:31
 Example 318:00
VI. Linear Transformations
Linear Mappings Revisited
24m 5s
 Intro0:00
 Linear Mappings2:08
 Definition2:09
 Linear Operator7:36
 Projection8:48
 Dilation9:40
 Contraction10:07
 Reflection10:26
 Rotation11:06
 Example 113:00
 Theorem 118:16
 Theorem 219:20
Kernel and Range of a Linear Map, Part I
26m 38s
 Intro0:00
 Kernel and Range of a Linear Map0:28
 Definition 10:29
 Example 14:36
 Example 28:12
 Definition 210:34
 Example 313:34
 Theorem 116:01
 Theorem 218:26
 Definition 321:11
 Theorem 324:28
Kernel and Range of a Linear Map, Part II
25m 54s
 Intro0:00
 Kernel and Range of a Linear Map1:39
 Theorem 11:40
 Example 1: Part A2:32
 Example 1: Part B8:12
 Example 1: Part C13:11
 Example 1: Part D14:55
 Theorem 216:50
 Theorem 323:00
Matrix of a Linear Map
33m 21s
 Intro0:00
 Matrix of a Linear Map0:11
 Theorem 11:24
 Procedure for Computing to Matrix: Step 17:10
 Procedure for Computing to Matrix: Step 28:58
 Procedure for Computing to Matrix: Step 39:50
 Matrix of a Linear Map: Property10:41
 Example 114:07
 Example 218:12
 Example 324:31
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For more information, please see full course syllabus of Linear Algebra
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1 answer
Last reply by: Professor Hovasapian
Wed Oct 26, 2016 7:58 PM
Post by Kaye Lim on September 23, 2016
For example 1, If I choose a specific number for the random vector (a,b,c), then I will get a specific number value for (c1,c2,c3). Because the solution of (c1,c2,c3) exists, we conclude that the given 3 vectors v1,v2 and v3 span the vector space R^3.
However, for example 4, we got infinite number of solution for (x1,x2,x3,x4). I thought we would conclude that the set of 4 given vectors (1,2,1,4),(1,2,1,4),(0,1,1,1) and (2,5,3,9) would span the Null space as in example 1. Why in example 4, the solution vectors span the Null space instead of the 4 given vectors?
1 answer
Last reply by: Professor Hovasapian
Wed Sep 2, 2015 11:44 PM
Post by Alexander Tetreault on August 31, 2015
Hi Raffi,
I am somewhat confused by the definition of 'span', it seems as though it has two meanings. In the subspaces video you defined it as the
"set of all linear combinations of the elements in S" while in this video it was defined as being the set of vectors by which all other vectors are a linear combination of. Basically, by my understanding, one means the set of elements created while the other means the set of vectors that do the creating. Could you please clarify?
2 answers
Last reply by: Growth Mindset Believer
Sun Apr 3, 2016 11:44 PM
Post by Growth Mindset Believer on May 19, 2015
This is not a question. I just wanted to say thank you for your lectures. I've gained so much from watching them. Often in mathematics I can understand how to do something computationally without understanding the underlying meaning of what I'm doing. However, after watching your videos multiple times I've gained a much deeper understanding of linear algebra, which is helping me a great deal with the linear algebra course that I'm currently enrolled in.
What I usually do is watch your lecture before reading the section of the book that I'm working on. Then, I try a few examples and go back and watch your lecture again to gain a deeper understanding of what I've done. Maybe some students only need to hear something once in order to fully understand it, but I've found that repetition of concepts and regular practice tend to be the only ways in which I can get an A in a course, especially as I've gone higher in mathematics and the material has gotten more complicated.
This is why online lectures such as yours are so great since I can watch them however many times as I want whereas any instructors will grow tired of explaining something multiple times in person, and I tend to worry that I'm holding back the rest of the class from learning if I ask too many questions in a real life lecture hall since time is limited and the professor has to get through his or her lesson plan. These issues do not exist with online lectures; if I don't understand a concept, I just watch it again until I get it.
This is also part of the beauty of mathematics, since going over past material can shed light on new concepts that I didn't understand the first time around, so it's like a well that I can continually draw water from. I also prefer your lectures over the ones I find on youtube since you are a trained mathematician who has taught higher education courses so your teaching style is refined and your lectures are well structured, as opposed to a lot of the videos I'll find on youtube where the person doesn't really have a good grasp of the material and just skips to computation without explaining what they are doing.
Lastly, I appreciate your personality. You come across as a kind uncle type; I can't detect a hint of arrogance in your personality as opposed to many of my real life professors who act like the students are beneath them. Thank you again for making your videos and sorry for writing such a long post, I just wanted to thank you for helping me with my linear algebra course this semester.
2 answers
Last reply by: Christian Fischer
Tue Oct 1, 2013 2:24 AM
Post by Christian Fischer on September 25, 2013
Hi Raffi: Just a question for example 4. Is it correctly understood that Since we have 2 free parameters "s" and "t" this does NOT mean we have a infinate number of vectors in our nullspace because t*(1,1,0,0) (our solution vector) is the same vector no matter what value of t we use (it has the same direction and can just be scaled up and down)??
So i mean (1,1,0,0) is the same vector as (5,5,0,0)?
4 answers
Last reply by: Professor Hovasapian
Wed Sep 25, 2013 4:57 PM
Post by Manfred Berger on June 15, 2013
I have a question regarding example 2: Since P_1(t) and P_2(t) are both second degree polynomials,there are no vectors present in this set to span any vector below degree 2. Isn't it therefor obvious that this can't be a spaning set for the entire space even before checking it formally?
2 answers
Last reply by: Professor Hovasapian
Tue Feb 19, 2013 12:43 AM
Post by Megan Kell on February 17, 2013
at 21:00 you say that the only way this solution could be consistent is if b4a+2c = 0, and the only way that this is possible is if b, a, and c are all zero, and since that would be a trivial solution, this system is inconsistent and therefore has no solution. Why can't b=0, a=1 and c=2? This would also cause b4a+2c = 04(1)+2(2)=0 and it would not be a trivial solution, thus allowing the system to be consistent. Why is this not possible?