Is this is linear transformation?

In summary, the question involves a linear transformation from R2 to R3, with the function F mapping (x1, x2) to (x1, x2, 0). The definition of linear transformation is needed to solve the problem, and the attempt at a solution suggests that F(x) could be equal to Ix, where I is the identity matrix. To determine linearity, the student must check whether vector addition and scalar multiplication are preserved.
  • #1
Jamin2112
986
12

Homework Statement



My buddy was asking me this question. It's from his linear algebra homework.

F: R2 --> R3
F[(x1 x2)] = (x1 x2 0)

Homework Equations



I can't remember the definition of "linear transformation." Hopefully it's not too complicated.

The Attempt at a Solution



I don't think you have a vector with 2 rows be transformed into a vector with 3 rows. However, (x1 x2 0) really is just equal to (x1 x2), so we could have F(x)=Ix, where I is the identity matrix. (Right? I dunno. I haven't taken this class for quite some time)
 
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  • #2
To determine whether this is linear, your buddy needs to check whether it preserves vector addition and scalar multiplication.
 

FAQ: Is this is linear transformation?

Is this function a linear transformation?

A linear transformation is a mathematical function that preserves vector addition and scalar multiplication. To determine if a function is a linear transformation, you can check if it satisfies the properties of linearity, which include additivity and homogeneity. If the function satisfies both properties, then it is a linear transformation.

How can I tell if a matrix represents a linear transformation?

A matrix represents a linear transformation if its columns form a linearly independent set. This means that none of the columns can be written as a linear combination of the other columns. Additionally, the number of rows in the matrix should be equal to the number of dimensions in the vector space. If these conditions are met, then the matrix represents a linear transformation.

What is the difference between a linear transformation and a non-linear transformation?

A linear transformation preserves the properties of linearity, such as additivity and homogeneity, while a non-linear transformation does not. This means that a linear transformation will map straight lines to straight lines, while a non-linear transformation may map them to curves or other shapes.

Can a linear transformation have a constant term?

No, a linear transformation cannot have a constant term. This is because a linear transformation must satisfy the property of homogeneity, which means that multiplying the input by a scalar will result in the output being multiplied by the same scalar. A constant term would violate this property.

Are all linear transformations invertible?

No, not all linear transformations are invertible. For a linear transformation to be invertible, it must be both one-to-one and onto. This means that each input must map to a unique output, and every output must have at least one corresponding input. If a linear transformation does not satisfy these conditions, then it is not invertible.

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