# Linear Algebra Subspaces

Section 4: Real Vector Spaces: Lecture 2 | 43:37 min

Continuing our lesson on vector spaces will lead us to the notion of vector subspaces and linear combinations. Essentially what you’ll be dealing with here is more rule-checking like we did last time. A space can only be a subspace if it follows specific rules set by the vector space. Things may seem confusing now, but once you apply this knowledge to some examples you’ll quickly develop a way to view spaces/subspaces in your own way. Additionally, we’ll discuss combining vectors in a space, and what applications that new vector has. After this we’ll get a breather from the technicalities of proofs, and jump into a topic with more math than words.

Table of Contents

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

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

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

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

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 Row-Equivalent 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

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

40m 10s

- Intro0:00
- Finding the Inverse of a Matrix0:41
- Finding the Inverse of a Matrix0:42
- Properties of Non-Singular 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 Non-Singular Equivalences28:37
- Example: Does the Following System Have a Non-trivial Solution?30:13
- Example: Inverse of a Matrix36:16

21m 25s

- Intro0:00
- Determinants0:37
- Introduction to Determinants0:38
- Example6:12
- Properties9:00
- Properties 1-59:01
- Example10:14
- Properties, cont.12:28
- Properties 6 & 712:29
- Example14:14
- Properties, cont.18:34
- Properties 8 & 918:35
- Example19:21

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 Non-Singular Equivalences43:07
- List of Non-Singular Equivalences43:08
- Example44:38
- Cramer's Rule52:22
- Introduction to Cramer's Rule and Example52:23

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

52m 44s

- Intro0:00
- n-Vectors0:58
- 4-Vector0:59
- 7-Vector1:50
- 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
- Cauchy-Schwarz Inequality24:56
- Triangle Inequality36:29
- Unit Vector40:34
- Vectors and Dot Products44:23
- Orthogonal Vectors44:24
- Cauchy-Schwarz Inequality45:04
- Triangle Inequality45:21
- Example 145:40
- Example 248:16

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

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

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

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

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

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

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

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

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 Non-Homogeneous Systems19:47

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

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 Non-Singular Equivalences24:24
- List of Non-Singular Equivalences24:25

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

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

32m 53s

- Intro0:00
- Orthonormal Bases in n-Space1:02
- Orthonormal Bases in n-Space: 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

21m 27s

- Intro0:00
- Orthogonal Complements0:19
- Definition0:20
- Theorem 15:36
- Example 16:58
- Theorem 213:26
- Theorem 315:06
- Example 218:20

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

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

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

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

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

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

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

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

For more information, please see full course syllabus of Linear Algebra

1 answer

Last reply by: Professor Hovasapian

Sat Apr 12, 2014 5:24 PM

Post by Kasun Jayasuriya on April 7, 2014

Wow. These lectures and examples are really good. I have just one month for my exams and I haven't understood any of this till now because university lectures are completely useless. I am really glad that I found this at least just before my exams. This really helps. Thanks a lot.

0 answers

Post by Burhan Akram on November 8, 2013

Hello Professor Raffi,

Very good explanation of subspaces. However, I have a question about one subspace. How would you approach this to show whether it's a subspace or not. Here is it, " subset of all polynomials in P5 for which p(1)=p(3)"....how would you tackle this problem? Thanks again, your lectures are very helpful.

4 answers

Last reply by: Professor Hovasapian

Wed Sep 25, 2013 4:19 PM

Post by Christian Fischer on September 24, 2013

Hi Raffi, Once again: Great video. I want to make sure I understood the conclusion to example 2 100%. Can you clarify that this is correct?

So w is a subspace of the Vector space V because:

A)The subset (a,b,0) satisfies the list of operations for vector spaces (closure etc.) when applying the 2 operations (+) and (*)

B)When testing for closure the vector A1+A2=(a1+a2,b1+b2,0) Is of the same form, so if we ended up with a non-zero z-cordinate it would be outside of the subset and therefore the subspace is not a vectorspace.

Can you say R^3 alone is a vector space? Does it mean that every vector in R^3 is a vector space? And so R^2 is a subset of R^3 but not vice versa.

Have a great day

Christian

0 answers

Post by Professor Hovasapian on June 18, 2013

Hi Manfred,

Thank you so much for the kind words -- they truly mean a lot to me. I'm thrilled that Mathematics is beautiful to you; and I'm happy we can bring some of this beauty to you in our courses and lessons.

Have fun.

Raffi

0 answers

Post by Manfred Berger on June 15, 2013

Just when I thought the lecture was losing a bit of momentum, you pull a VS from the polynomial ring, and have my full attention back in an instant. You probably get messages like this all the time, but: You're just an awesome instructor!

4 answers

Last reply by: Professor Hovasapian

Sun Mar 24, 2013 6:10 PM

Post by Professor Hovasapian on March 23, 2013

Hi Matt,

I you're doing well.

When a set is closed under addition, this means that if I take two elements from that set and add them, the result is yet another element in that set. For example, the set of even numbers: if I take any two even numbers and add them, the result is always an even number -- so the even numbers are closed under addition. Now let's take the odd numbers: if I add any two odd numbers, I get an even number -- the result of the addition lands me outside the set of add numbers -- so the odd numbers are NOT closed under addition.

Now, this question. Our set is given, and the two elements are the vectors (a, a-2) & (b,b-2). When I add these I get the (a+b, a+b-4). Does it make sense how we got this?

The question is: Is this result vector in the original set? The first component is a+b. Let's call this M. The second component is a+b-4. Let's call this M-4. Now we have (M,M-4).

Does this last vector look like it belongs to the original set? NO -- because any vector in the original set has to be of the form (P, P-2). When we added we got (m, M-4) -- we landed outside of the set -- therefore NOT closed under addition...therefore, NOT a Subspace.

I hope this made sense. Please let me know if it did not, and I will prepare a short document for you with other examples and upload it to my "Linear Algebra for Educator.com" Group page on Facebook:

https://www.facebook.com/groups/344583348957004/

Best wishes, and take good care

Raffi

1 answer

Last reply by: Professor Hovasapian

Sat Mar 23, 2013 5:16 PM

Post by Matt Cypert on March 23, 2013

Hello,

V=R^2. If I had a subset S = {(x,x-2): xâˆˆR} is it a subspace of V.

First I use addition

u = (a, a-2)

v = (b, b-2)

u+v = (a+b, a + b - 4). The book says it is not closed under addition, therefore it is not a subspace, but it does not go into any detail as to why. In all your examples they seemed to all work out or be a subspace. Could you possibly explain to me why this is not closed under addition making it not a subspace?

1 answer

Last reply by: Professor Hovasapian

Wed Jan 23, 2013 3:52 AM

Post by Hai Lieu on January 22, 2013

Hello Pro. Professor Hovasapian,

Thanks for a quick respond for the above question. So, if a 2x2 matrix A is given. How do we find A-invariant subspaces?

Thanks

Hai

2 answers

Last reply by: Professor Hovasapian

Tue Jan 22, 2013 9:27 PM

Post by Hai Lieu on January 22, 2013

Does anyone know the different between A-invariant subspace and subspace? I don't see any lesson regarding A-invariant subspace.

Thank you in advance

1 answer

Last reply by: Professor Hovasapian

Tue Nov 20, 2012 6:52 PM

Post by Brodey Hansen on November 20, 2012

My professor should watch these lectures so he can learn how to TEACH!

1 answer

Last reply by: Professor Hovasapian

Sun Aug 5, 2012 7:52 PM

Post by hasan dilek on August 5, 2012

great lectures, good examples. it feels like the lectures at uni are completely useless.

0 answers

Post by Talwar Chanonia on November 18, 2011

Thankyou!!