Linear Algebra Transformations

In summary, to find the image of the vectors 1, t, and t2 under the transformation T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2, you substitute the coefficients of the given vectors into the transformation equation and simplify to get the resulting images: T(1) = 3+5t T(t) = -2t+4t2 T(t2) = t2
  • #1
henry3369
194
0

Homework Statement


T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2
Find the image of the vectors :
1. 1
2. t
3. t2

Homework Equations


T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2

The Attempt at a Solution


I don't know how my book solves these transformations, but the answers are:
T(1) = 3+5t
T(t) = -2t+4t2
T(t2) = t2

How do you substitute a single vector for an entire expression to solve for each of these?
When it was a simple transformation (T(x) = x^2), you just replace x with the input, but for this one, you have to substitute an entire expression to find the transformation.
 
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  • #2
henry3369 said:

Homework Statement


T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2
Find the image of the vectors :
1. 1
2. t
3. t2
This is just a matter of understanding and applying the given definition (and arithmetic).
You are told that T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2
and asked to find T(1). 1= 1+ 0x+ 0x2. a0= 1, a1= 0 and a2= 0
T(1+ 0x+ 0x2)= 3(1)+ (5(1)- 2(0))t+ (4(0)+ 0)t2= 3+ 5t.

Similarly, t= 0+ 1t+ 0t2 so a0= 0, a1= 1, and a2= 0.

t2= 0+ 0t+ 1t2 so a0= 0, a1= 0, and a2= 1.
2. Homework Equations
T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2

The Attempt at a Solution


I don't know how my book solves these transformations, but the answers are:
T(1) = 3+5t
T(t) = -2t+4t2
T(t2) = t2

How do you substitute a single vector for an entire expression to solve for each of these?
When it was a simple transformation (T(x) = x^2), you just replace x with the input, but for this one, you have to substitute an entire expression to find the transformation.
 

FAQ: Linear Algebra Transformations

What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the structure of the vector space. This means that the linear transformation preserves addition and scalar multiplication operations.

What is the difference between a linear transformation and a linear map?

There is no difference between a linear transformation and a linear map. These terms are used interchangeably to refer to the same concept.

What are some examples of linear transformations?

Examples of linear transformations include rotations, reflections, dilations, and shears. These are all transformations that preserve the structure of the vector space, such as the length and direction of vectors.

What is the matrix representation of a linear transformation?

The matrix representation of a linear transformation is a matrix that represents the transformation in terms of its effects on the standard basis vectors of the vector space. This matrix can be used to perform the same transformation on any vector in the vector space.

How is linear algebra used in real life?

Linear algebra is used in a variety of fields, such as computer graphics, engineering, economics, and physics. It is used to solve systems of equations, analyze data, and model real-world situations. For example, linear transformations are used in computer graphics to rotate or scale objects, and in economics to analyze supply and demand relationships.

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