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jinksys
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Homework Statement
Show that 2.1.1 is equivalent to the totality of 2.1.2 and 2.1.3.
Homework Equations
The Attempt at a Solution
aTx + bTy = aT(x) + bT(y) = T(ax) + T(by) = T(ax + by) ?
jinksys said:I can 'get' that 2.1.1 follows from 2.1.2 and 2.1.3 and visa versa, but I'm not sure how to say it as a proof. I've never been good with proofs.
Dick said:You aren't bothered by (2.1.3) saying T(ax)=a*x?? You should be. But if you aren't good at proofs, maybe you aren't. Change (2.1.3) to T(ax)=a*T(x). The first stage of the proof is show that (2.1.1) implies (2.1.2) and (2.1.3). That's pretty easy. I'll give you a hint for the first part. Put a=1 and b=1 into (2.1.1). What do you conclude?
jinksys said:I meant to say that I understood that T(ax)=aT(x), but only because I have other linear algebra books to help me get through this class.
If a=b=1,
T(ax +by) = T(1x + 1y) = T(1x) + T(1y) = 1T(x) + 1T(y) = T(x) + T(y)
jinksys said:I'm not sure if you saw my edit or not, so I'll just give it its own post.
To show 2.1.3, do I let a=a and b=0?
T(ax + by) where a=a, b=0:
T(ax + 0y) = T(ax) = aT(x).
Conversely with a=0, b=b:
T(0x + by) = T(by) = bT(y)
jinksys said:If you let s = ax and t = by,
then T(s + t) = T(s) + T(t) = T(ax) + T(by) = aTx + bTy,
So, T(ax + by) = aTx + bTy ?
Any closer?
jinksys said:So letting s=ax and t=by was the right thing to do?
Are there other ways to say 2.1.1 stems from 2.1.2 and 2.1.3?
A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the original space. In simpler terms, it is a function that takes in a vector and outputs another vector, while following specific rules.
There are three main properties of a linear transformation. First, it must follow the rule of additivity, meaning that the transformation of the sum of two vectors is equal to the sum of their individual transformations. Second, it must follow the rule of homogeneity, meaning that the transformation of a vector multiplied by a scalar is equal to the scalar multiplied by the transformation of the vector. Lastly, it must preserve the zero vector, meaning that the transformation of the zero vector is equal to the zero vector.
To prove that a function is a linear transformation, you must show that it follows the three properties mentioned above. This can be done by using algebraic manipulation to show that the function satisfies the additivity and homogeneity properties. Additionally, you can show that the function preserves the zero vector by substituting in the zero vector and showing that it remains unchanged.
Matrices are a useful tool for representing and performing linear transformations. Each linear transformation can be represented by a unique matrix, and by multiplying this matrix with a vector, the transformation can be applied to that vector. This makes it easier to visualize and understand how a linear transformation affects vectors in a vector space.
No, not all functions can be considered linear transformations. For a function to be a linear transformation, it must follow the three properties mentioned earlier. If a function does not satisfy these properties, then it cannot be considered a linear transformation. Additionally, a linear transformation must be a mapping between two vector spaces, so functions that do not fit this criteria cannot be considered linear transformations.