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# Sums and scalar multiples of linear transformations

Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication. Created by Sal Khan.

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• Why would we add two Transformations

Scaling is fine, its like transform and scale the resultant or scale the transformation matrix itself and transform.

If I am gonna apply two transformation one after another, that would be S(T(x)) = (S(T))(x) = A * (B * X) = (A * B) * X. So I would need to multiply matrices to chain transformations.

I can say there is Transformation F that is defined by Sum of Transformations F = S + T, so we do like above.

But whats the intuition?
• Multiplying is saying transform vector c into vector y, then transform vector y into vector z, so you get a different vector in between.

Adding meanwhile is saying do both transformations to vector x for a new vector y.

Of course both have rules they have to follow. in addition the matrices need to have the same dimensions and in multiplication the number of columns of the leftmost matrix needs the same number of rows for the matrix to the right of it.

So both have their uses and limitations.
• Why Sal defined them are definitions...Are just results of the distributing property of matrix multiplication?
• Both of these definitions are similar to the distributive property of matrix multiplication but they more than just that: 1) S & T are more than just matrices, they are linear transformations; and, 2) a scalar is not a matrix. So as you can neither really uses the distributive property of matrix multiplication.

That's why they get their own definitions.
• So to add the transforms, they need to both be mappings from R^n to R^m? Or another way of saying it: their matrices have to have the same dimensions to add the transforms?
• Yes, to add two matrices they have to have the same number of rows and columns (the same dimension).
(1 vote)
• When might you want to add two transformations?
• (cS)(x) = c(S(x))

What is cS? S is a transform right? It also a linear transform. Ok.

Lets say we use function notation.

S: R^n -> R^m

What is S a member of? What is c*S a member of? R^n, R^m?

I am lost. Did I miss the part where we talked about sets of functions?
(1 vote)
• S is a linear transformation that maps elements from R^n to R^m
cS is also a linear transformation that maps elements from R^n to R^m
• What does (cS) (x) do? Is it "apply S to x c times"? Is each consecutive transformation applied to the initial x, and they´re all summed up?
(1 vote)
• Good question. Now I understand why it has to be defined. The definition is:

c(S(x)).

Which means: transform the vector "x" (just once) with the transformation "S", then multiply the result by "c"
• this is confusuing nay tips and or pointers
(1 vote)
• Try proving these things for yourself before watching Sal do it. Try anything you can think of.
• Question, at about the mark, Sal creates a definition, "S(x) =A(x), T(x)=B(x) (x has vector notation). he then creates a matrix stating, A=[a1,a2. . .an] (the a's have vector notation). conversely, he does the same with "B". He then multiplies the two vectors, "Ax" which is a dot product a1x1+a2x2+. . .anxn, however this time the a's have a vector notation, the x's do not have the vector notation (line on top). I realize that dot products are scalars, however, why the insistence that while both are vectors, one stops becoming a vector?
(1 vote)
• I wouldn't call Ax a dot product. A dot product takes two vectors and returns a number. Ax can be thought of taking a vector in Rn to Rm. So Ax outputs a vector. Remember x is vector [x1, x2,... xn] where the components are real numbers. So Ax is just the linear combination of the column vectors of A (a1, a2,...... an) where the coefficients are the components of x , that is Ax = x1a1+x2a2+.....+xnan.
• What is the difference between (s+t) and (A+B)?
(1 vote)
• You mean "S + T"? The transformations are, I think, "S(x) = Ax" and "T(x) = Bx" "A" and "B" aren't transformations, they're the matrices of the transformations.

We say "f(x) = 2x" and we say the function is "f", not "2". We might say that f(x) is 2x or x^2 or sin(x), Do we say that the function is "sin"? "2"? "^2"?

I'm not sure that logically it needs to be this way, but this is how it's done.
(1 vote)
• if two vectors are scalar multiples of one another, are they parallel?
(1 vote)