# Linear transformation with standard basis

• foreverdream
In summary, the conversation discusses finding the matrix of a linear transformation from P2 to R3 and the matrix of the transformation with respect to the standard bases for each. The process involves taking a vector in P2, applying the transformation, and expressing the result using R3's basis. This matrix acts as a "translator" between the two spaces.
foreverdream

## Homework Statement

Let s be the linear transformation
s: P2→ R^3 ( P2 is polynomial of degree 2 or less)
a+bx→(a,b,a+b)
find the matrix of s and the matrix of tos with respect to the standard basis for the domain
P2 and the standard basis for the codomain R^3

## The Attempt at a Solution

Now I know that Standard basis of P2 is {1,x}
and standard basis for R^3 = {(1,0,0), (0,1,0), (0,0,1)}
and s(1)= (1,0,1)
and s(x) = (0,1,1)

but I don't know how to proceed from here?

Well we know s: P2 -> R3 so the map of s will have 2 columns and 3 rows. The first row will be (1, 0). Can you do the rest?

To find the matrix with respect to P2's basis and R3's basis, you proceed by taking a vector in P2, applying s to it, then expressing the result using R3's basis. Repeating this for each vector in P2 gives the matrix. How do these compare?

You might find this helpful: http://www.millersville.edu/~bikenaga/linear-algebra/matrix-linear-trans/matrix-linear-trans.html .

It might be helpful to think of the resultant matrix as a way of "translating" between the languages of P2 and R3.

Hope that helps!

Last edited by a moderator:
Thanks. I'll try and post back my findings.

am I correct?

Last edited:
still working on it

Last edited:
trying not suceeded

i resolved it no help needed

## 1. What is a linear transformation?

A linear transformation is a mathematical operation that maps one vector space to another in a linear manner. It involves multiplying each element of a vector by a constant and then adding them together to create a new vector.

## 2. What is a standard basis?

A standard basis is a set of vectors that form the basis for a given vector space. These vectors are typically orthogonal (perpendicular) and have a length of 1. In a two-dimensional space, the standard basis consists of two vectors, (1, 0) and (0, 1).

## 3. How do you perform a linear transformation with standard basis?

To perform a linear transformation with standard basis, you need to multiply each element of the vector by the corresponding element in the standard basis vector. For example, if the standard basis is (1, 0) and (0, 1), and the vector is (2, 3), the resulting transformation would be (2, 3).

## 4. What is the purpose of using standard basis in linear transformations?

The standard basis is used in linear transformations because it simplifies the calculations and makes it easier to understand and visualize the transformation. It also allows for a more efficient representation of vectors and matrices.

## 5. Can you explain the concept of a linear transformation with an example?

Yes, for example, consider the transformation T(x,y) = (2x + y, x - y). This transformation maps a vector (x,y) to a new vector (2x + y, x - y). If we apply this transformation to the standard basis vectors, we get (2(1) + 0, 1 - 0) = (2, 1) and (2(0) + 1, 0 - 1) = (1, -1). These two resulting vectors form the new standard basis for the transformed vector space.

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