Functions and linear transformations
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A more formal understanding of functions
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Vector Transformations
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Linear Transformations
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Matrix Vector Products as Linear Transformations
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Linear Transformations as Matrix Vector Products
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Image of a subset under a transformation
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im(T): Image of a Transformation
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Preimage of a set
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Preimage and Kernel Example
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Sums and Scalar Multiples of Linear Transformations
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More on Matrix Addition and Scalar Multiplication
More on Matrix Addition and Scalar Multiplication More on Matrix Addition and Scalar Multiplication
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- In the last video we started off with two linear
- transformations.
- We had the linear transformation s that was a
- mapping from Rn to Rm.
- And then we had the linear transformation t, that was
- also a mapping from Rn to Rm.
- And we defined the idea of the addition of these two
- transformations.
- So s plus t, this transformation of x we defined
- as being equal to s of x, this vector plus t of x.
- And of course, this input is still from Rn, and then each
- of these are vectors in Rm.
- If we add two vectors in Rm to each other, we get another
- vector in Rm because Rm is a valid subspace.
- It's also closed under addition.
- So this is still a mapping.
- So s plus t is still a mapping from Rn to Rm.
- And we also said that every linear transformation we've
- shown in a previous video, can be represented as a matrix.
- We could say that s of x is equal to some
- matrix a times x.
- And we could also say that t of x is equal to some
- matrix b times x.
- And both of these would be m by n matrices.
- And let me write that m by n, both of these.
- Because these are both mappings from Rn to Rm.
- And what we did is we made a another definition.
- This was a definition right here and then we made another
- definition.
- We defined the addition of two matrices.
- We said any matrix a plus b, they both have to have the
- same dimensions.
- So they're both m by n in this case.
- And we defined this addition to be a new matrix, where each
- column of this matrix is the sum of the corresponding
- columns of these matrices.
- So this matrix's first column will be the sum of a's first
- column and b's first column.
- So a1 plus b1, the second column I'll do a little line
- here is, a2 plus b2.
- And it goes all the way to An plus Bn.
- This was a definition.
- And the whole reason why we made this definition, is
- because if you defined matrix addition in this way, then
- this thing, when you replace it with Ax plus Bx, you get to
- that this thing over here is equal to the corresponding
- matrices by this definition, a plus b times x.
- This was the motivation to get to a nice expression, like
- this, for defining matrix addition in this way.
- Now this all seems very abstract, so let's actually
- add a matrix, or let's add two matrices.
- So we'll start off with a two-by-two case.
- So let's say I'm adding the matrix 1, 3, minus 2, 4 to the
- matrix, remember they have to have the same dimensions, to
- the matrix 2, 7, minus 3, minus 1.
- What do we get?
- Well by definition, you just add up their
- corresponding columns.
- You add up the first column.
- When you add up the corresponding columns, what
- happens when you add up two columns, two vectors?
- Well, you just add up their corresponding entries.
- So essentially, when you add up to matrices, you're just
- adding up all of the corresponding entries.
- I'll talk about it in this way, just because that's how I
- defined it, but they're all equivalent.
- The first thing, the first column, in this matrix right
- here, is going to be this column plus this column.
- So it's going to be 1 plus 2, let me write it like this, and
- then minus 2, minus 3.
- And then the second column, that one right there, is going
- to be 3 plus 7 and then 4 minus 1.
- And so this will be equal to 3, 10, minus 5, and
- 3, just like that.
- And notice, even though the definition is I'm adding up
- corresponding columns.
- Well, what in effect happened?
- Well, I'm just adding up the corresponding entries.
- I added the 1 to the 2, the 3 to the 7, the minus 2 to the
- minus 3, the 4 to the minus 1.
- It's that straightforward.
- Nothing fancier than that.
- In fact, we could have rewritten this definition.
- If we say that the vector or the matrix a is equal to a11
- a12, all the way to a1n.
- And then if you go down this is a21 all the way to an1.
- And then you go all the way down there to ann.
- We've seen that before.
- And then the matrix b, just by the same argument or by
- similar definition, this is b11, that entry is b11, that's
- b12, all the way to b1n.
- This is b21, second row, all the way down to bn, sorry this
- is m, we have m rows, so this is mn.
- So this right here is bm1, this would be bm2, all the way
- down to this is bmn, right there.
- Want to be very careful, these are m by n matrices.
- So this row down here is the mth row in both cases.
- But we could redefine our matrix, or another way of
- stating this matrix addition definition, is to say, if I'm
- going to add a plus b, I'm just going to add up the
- corresponding entries.
- So this entry up here is going to be-- do it in a different
- color --it's going to be a11 plus b11 this one's going to
- be a21 plus b21 all the way down to am1 plus bm1.
- And then this is going to be, of course, a12 plus b12 all
- the way to a1n-- let me scroll over a little bit --all the
- way over to a1n plus b1n, and then all the way
- down to amn plus bmn.
- These are equivalent definitions.
- This takes a lot less space to write in and I felt
- comfortable doing it because we've already
- defined vector addition.
- But it essentially boils down to you just add up all the
- corresponding entries.
- That's all matrix addition is.
- It's probably one of the simplest things that you've
- seen in your recent mathematical experience.
- Now, matrix scalar multiplication,
- very similar idea.
- We defined scalar multiplication times a
- transformation of x to be equal to a scalar times the
- transformation of x.
- This was a definition.
- And we also defined scalar times some matrix a to be
- equal to the scalar.
- A new matrix where each of its columns are the scalar times
- the column vectors of a.
- So a1, and then the next column is ca2, and then you go
- all the way to can.
- And the whole motivation for this was, because this could
- be simplified to-- well t I've said was equal to Bx, a times
- the transformation of x --the transformation t
- of x was equal to.
- So we still have our c.
- So it's going to be c times the matrix B,
- times the vector x.
- That's what the transformation of x could be written as.
- And so this would be equal to by just manipulating-- and we
- did this in the last video by actually breaking this up in
- the column vectors multiplying them by each of the components
- of x, and then distributing the c and rearranging them a
- little bit.
- We can now say, using this definition, that this is equal
- to some new matrix cB.
- We're using this definition, some new matrix cB, where it's
- essentially c times each of the column
- vectors of B times x.
- This right here was our motivation.
- We wanted to be able express this as a product of a some new
- matrix and a vector, because any linear transformation
- should be expressible in that way.
- And that's why we made this definition.
- Now let's apply it.
- And I wanted to show you that this is perhaps even simpler
- than matrix addition.
- So if we want to multiply the scalar 5 times the matrix,
- I'll do a 3 by 2 matrix.
- So 1, minus 1, 2, 3, 7, 0.
- This will just be equal to-- by this definition I'm just
- saying, I'm multiplying the scalar times each of the
- column vectors.
- So it's 5 times 1, 2, 7.
- But what's that?
- That's just 5 times each of the entries.
- It's 5 times 1, which is 5.
- 5 times 2, which is 10.
- 5 times 7, which is 35.
- Then the next column is going to be 5 times this column
- right here, which is just 5 times each of its entries.
- So 5 times minus 1 is minus 5.
- 5 times 3 is 15.
- 5 times 0 is 0.
- It's that simple.
- You literally, if we go back to this definition, we can
- define scalar multiplication of a matrix.
- So we could also define cA as being equal to, if we'd say
- this is a representation for A, of the scalar c times each
- of the entries of A.
- That's it.
- So it's c times a11, c times a12 all the
- way to c times a1n.
- You go down this way, c times a21 all the way down to c
- times am1 and then if you go down the diagonal,
- it's be c times amn.
- You literally just take your scalar and multiply it times
- every entry in A.
- And that's all you have to do.
- So hopefully this clarified things up a little bit, or
- maybe it was a bit of a review from things you might have
- learned in highschool.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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