One-to-one linear transformations

In summary, a linear transformation T(x)=Ax is one-to-one if and only if the columns of A are linearly independent. This is because the rank-nullity theorem states that T is one-to-one if and only if T(v) = 0 implies v = 0, which means that the columns of A must be linearly independent in order for the vector of coefficients (v) to be zero.
  • #1
Nikitin
735
27
Why is a linear transformation T(x)=Ax one-to-one if and only if the columns of A are linearly independent?

I don't get it...
 
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  • #3
Is there no alternative to insanely difficult wikipedia proofs?
 
  • #4
What does your textbook say? What is your textbook?

Do they prove the rank-nullity theorem??
 
  • #5
T is one-to-one if and only if T(x) = T(y) implies x = y, if and only if T(x-y) = 0 implies x - y = 0, if and only if T(v) = 0 implies v = 0. But T(v) is a linear combination of the columns of A, so this says the only way to combine the columns of A to get zero is if the vector of coefficients (v) is zero. In other words, the columns of A are linearly independent.
 
  • #6
Ahh, thanks Jbunny. It makes perfect sense now!

Micromass: Yes, it was, but the proofs in my book are written similarly to wikipedia - very tiresomely.
 

Related to One-to-one linear transformations

1. What is a one-to-one linear transformation?

A one-to-one linear transformation is a mathematical function that maps one set of data points onto another set in a way that preserves the underlying linear relationships between the points. This means that for every input value, there is only one corresponding output value and the relationship between the two sets remains consistent.

2. How is a one-to-one linear transformation different from other types of transformations?

Unlike other types of transformations, such as one-to-many or many-to-one, a one-to-one linear transformation has a unique relationship between the input and output values. This means that every input value has only one corresponding output value and vice versa.

3. What are some real-world applications of one-to-one linear transformations?

One-to-one linear transformations have many practical applications in areas such as economics, engineering, and data analysis. For example, they can be used to model the relationship between supply and demand in an economic system, or to analyze the correlation between different variables in a scientific study.

4. How can one-to-one linear transformations be represented mathematically?

One-to-one linear transformations can be represented using a matrix or a set of equations. For example, a 2D transformation could be represented by the equation y = mx + b, where m is the slope and b is the y-intercept. In matrix form, it could be represented as [y; x] = [m b; 0 1][x; y].

5. What is the importance of one-to-one linear transformations in data analysis?

One-to-one linear transformations are essential in data analysis because they allow us to accurately model and understand the relationships between variables. By transforming data into a linear form, we can more easily visualize and interpret the data, making it a powerful tool for analyzing and making predictions based on data.

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