Subspaces and Span
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- Опубліковано 25 бер 2019
- Now that we know what vector spaces are, let's learn about subspaces. These are smaller spaces contained within a larger vector space that are themselves vector spaces.
Script by Howard Whittle
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Note to viewers:
The vector space V (as in the video) is also a subspace of itself.
Hence, S does not have to be strictly smaller than V, as Dave slightly misleadingly stated in the introduction.
Awesone very nice explanation
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TY professor!
5:12 what if we multiplied by a negative scalar? Would we still get a matrix in the specified form?
Can you please explain what you meant by 'Any sum of these elements" in 3:20
I think it's important to note that a subspace must also contain the additive identity. In the case of vectors, it must contain the zero vector. Great video!
Very nice,Thx
the span of any number of elements of vector V is also a subspace of V
a span is the smallest subspace of V that contains this set of elements
span is important for describing vector spaces
Hey sir! I was just wondering, can the scalar for the 1st rule of Vector Spaces be a negative? If yes, wouldn't it make the matrix in the 1st question of Checking Comprehension not a subspace? Since the -b in the bottom row would turn positive
I'm not a professor but if you are talking about the 2bd question in last then if 2nd row if b becomes positive then b in first row will to -b thus form will remain same
thanK you so much.
I didnt understand the part where span of V is the smallest subspace of V. How come? The a1V1+a2V2+a3V3 (if linearly independent) is the entire R3 right?
For closure under addition, do the vectors that are added to vectors in a subspace have to be part of the subspace themselves?
yes
Thanks
What if c is negative?
exactly what I was thinking
Me too😅
Think it still works cause the bottom will be positive and the top will be negative, in other words, the bottom is the negative of the top line which is negative. A bit confusing but I think the rule he stated was that the bottom line is the negative of the top line, not that the bottom line itself is necessarily negative.
What are the difference between a Span and a Subspace?
Why the first question of the comprehensive is true? Could someone explain it please?
I have a midterm in 3 hours 😩 thank you so much
here is an interesting idea, since points in cartesian space are just sums of the i-hat and j-hat basis vectors with real coefficients technically speaking all of the 2-d coordinates system is simply span(i_hat,j_hat). Similarily the 3-d cartesian system is just span(i_hat,j_hat,k_hat)
I already miss your long hair
simon dx I agree
Sir, multiply vector x with any negative constant value. Then, will the resultant vector x belong to the set S?
same doubt
awesome
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English is my third language, and you still explain better than my professors in my mother tounge
i have a midterm tomorrow thx
2:20 is it really closed under scalar multiplication? what if c is negative???
No issue if c is negative. The original vectors can be any vectors in the form of [[x],[0],[-x]], where x is any real number. The multiplier c, can also be any real number. Multiplying any two real numbers together, also gets a real number, and x*c will still be the negative of -x*c.
PLEASE MAKE LECTURES ON REAL ANALYSIS
Thank you so much. By definition would a vector space be a (very useless) subspace of itself?
Know it's too late but for anyone with a similar question: V is in itself a subspace of V.
@@AEPPLE_MUSIC Makes sense
YES OFCOURSE! IT WOULD BE. 🙂
Sir in Example of subspace what if we take the value of scalar as negative then the 1st property will not be held . Please help me with my doubt.
How
in 5:09 (2), can their span be in real number instead of a & b?
well, you assume a and b to be constants that are also real numbers. so the span, as a result, will be a real number, as you're not dealing with any variables here...hope this helps :)
What if sub space doesn’t include identity O but satisfies closure .
It’s not a vector space is it? Still a subspace?
If so not every subspace is vector space. Am I missing something ?
Multiplying zero scalar to a vector will yield zero result,
So, in case of subspace, we could say that it is closed for scalar multiplication?
c = 0 makes me think of another question. If c = 0, then the vector is [0,0,0]. which means it's not maintaining the [x,0,-x] form??? idk. pls help
Great ! Thanx! 😂
2:00 why we are checking its closed or not, since its a subset of vector space..
Confused!
we are trying to check if it is indeed a subset , thats why
what if you had used -1 as a scalar to multiply?
i am facing the same issue
Every subspace of R5 that contains a nonzero vector must contain a line. Is this statement true?
yes!
Watching 10 min before exam
2:28 what if c was -1
Scalar positive integer
@@bigilpandi7722 wrong
c can be any real number, remember it's a scalar, so it can be negative. thus, it will still work as the first component of the x vector has the opposite sign of the third component of the x vector, so it still satisfies this closure property
@@multitude1337 so what does 'form' really mean? im confused
@@Christian-mn8dh Think of form as meaning pattern. A vector in the form of [[x],[0],[-x]] means that you can put any (real in this case) number in the position of the x, in both the first and final entry of this vector. So this means that [[4], [0], [-4]] as well as [[-6], [0], [-6]] are vectors of this form. They have something in common, in that their first and final entries are negatives of each other, and they have zero for the middle entry.
Note that the nested brackets is my way of indicating the vertical matrix, in an inline text description. Think of the innermost brackets as individual rows, and the outermost bracket as the full matrix of those rows. In this case, there's just one entry per row, since vectors in linear algebra are considered vertical matrices.
I don't understand R 2*2 Could you explained it
he made a video called understanding vector space
The R refers to real numbers. The 2x2 refers to 2x2 square matrices. Putting it together, it refers to the set of all square matrices with 2 rows and 2 columns that contain any real number in each of the 4 entries.
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2:09, why is it “-(cx)”?
c * (-x) is -cx
just to illustrate that it's some number in the form -x better
Multiplication of real numbers is associative and commutative, so you can rearrange the parentheses and negative sign, however you prefer.
Can someone explain #2 to me?
french: Span is written as Vect
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