Linear Transformations: Why w1 is a Linear Combination of v

In summary, the ability to write any coefficient of w, such as w1, as a linear combination of the coefficients of v is a result of the definition of linear transformations. This is possible because both w and v can be expressed in terms of coefficients using a basis, and this same basis can be used to express the linear transformation as a matrix. Therefore, the coefficients of v and w can be written as a linear combination using the same coefficients as the linear transformation.
  • #1
Amin2014
113
3
Given w = T (v), where T is a linear transformation and w and v are vectors, why is it that we can write any coefficient of w, such as w1 as a linear combination of the coefficients of v? i.e. w1 = av1 + bv2 + cv3

Supposably this is a consequence of the definition of linear transformations, but I don't see the connection.
 
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  • #2
In order to write ##w=(w_1,w_2,w_3)## and ##v=(v_1,v_2,v_3)## what do you need? And what does this mean for ##T##?
 
  • #3
fresh_42 said:
In order to write ##w=(w_1,w_2,w_3)## and ##v=(v_1,v_2,v_3)## what do you need? And what does this mean for ##T##?
We need a basis to be able to express a vector in terms of coefficents. I don't know what your second question implies.
 
  • #4
##T## is a linear transformation. What does this mean? Sure, it means that ##T(\alpha u + \beta v)=\alpha T(u)+\beta T(v)##, but since we have a basis, we can express this transformation in terms of the basis, too:
We write down ##T(1,0,\ldots ,0)\, , \, \ldots \, , \,T(0,\ldots,0,1)## as column vectors with respect to the same basis we used to write ##w##. The result is a number scheme which we call a matrix. Now it turns out that ##w=Tv## is exactly the matrix multiplication ##(w_1,\ldots,w_m) = \left( \sum_{j=1}^n T_{ij}v_j \right)_{1\leq i \leq m}## where ##n=\dim V## and ##m=\dim W##.

So the coefficients of ##v## and ##w## need a bases to write them as a tuple of components, and the same bases allow us to write the linear transformation as a number scheme, too.

Your example had ##n=m=3## and the coefficients are ##a=T_{11}, b=T_{12},c=T_{13}##.
 
  • #5
fresh_42 said:
##T## is a linear transformation. What does this mean? Sure, it means that ##T(\alpha u + \beta v)=\alpha T(u)+\beta T(v)##, but since we have a basis, we can express this transformation in terms of the basis, too:
We write down ##T(1,0,\ldots ,0)\, , \, \ldots \, , \,T(0,\ldots,0,1)## as column vectors with respect to the same basis we used to write ##w##. The result is a number scheme which we call a matrix. Now it turns out that ##w=Tv## is exactly the matrix multiplication ##(w_1,\ldots,w_m) = \left( \sum_{j=1}^n T_{ij}v_j \right)_{1\leq i \leq m}## where ##n=\dim V## and ##m=\dim W##.

So the coefficients of ##v## and ##w## need a bases to write them as a tuple of components, and the same bases allow us to write the linear transformation as a number scheme, too.

Your example had ##n=m=3## and the coefficients are ##a=T_{11}, b=T_{12},c=T_{13}##.
 
  • #6
Given ##T: V \rightarrow W ##, Consider a basis ## \{ v_1, v_2,...,v_n\} ## for ##V ##. Then ##T(v)=... ##
 
  • #7
fresh_42 said:
##T## is a linear transformation. What does this mean? Sure, it means that ##T(\alpha u + \beta v)=\alpha T(u)+\beta T(v)##, but since we have a basis, we can express this transformation in terms of the basis, too:
We write down ##T(1,0,\ldots ,0)\, , \, \ldots \, , \,T(0,\ldots,0,1)## as column vectors with respect to the same basis we used to write ##w##. The result is a number scheme which we call a matrix. Now it turns out that ##w=Tv## is exactly the matrix multiplication ##(w_1,\ldots,w_m) = \left( \sum_{j=1}^n T_{ij}v_j \right)_{1\leq i \leq m}## where ##n=\dim V## and ##m=\dim W##.

So the coefficients of ##v## and ##w## need a bases to write them as a tuple of components, and the same bases allow us to write the linear transformation as a number scheme, too.

Your example had ##n=m=3## and the coefficients are ##a=T_{11}, b=T_{12},c=T_{13}##.
Got it!
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the basic structure of the original space. In simpler terms, it is a transformation that maintains the properties of linearity, such as scaling and addition.

2. What is a linear combination?

A linear combination is a mathematical operation that combines two or more vectors by multiplying each vector by a scalar and then adding them together. In the context of linear transformations, it refers to the combination of two or more vectors to create a new vector.

3. How does a linear transformation relate to a linear combination?

A linear transformation can be thought of as a generalization of a linear combination. In a linear transformation, a vector is multiplied by a matrix to produce a new vector, whereas in a linear combination, two or more vectors are multiplied by scalars and then added together to create a new vector.

4. Why is w1 a linear combination of v in linear transformations?

In linear transformations, w1 is a linear combination of v because it can be expressed as a linear combination of the columns of the transformation matrix. This means that w1 can be obtained by multiplying each column of the matrix by a corresponding scalar and then adding them together.

5. How is the linearity of a transformation determined?

The linearity of a transformation can be determined by checking if it satisfies two properties: additivity and homogeneity. Additivity means that the transformation of a sum of vectors is equal to the sum of their transformations. Homogeneity means that the transformation of a vector multiplied by a scalar is equal to the scalar multiplied by the transformation of the vector.

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