Matrix condition for one-to-one trans | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

A detailed explanation of the null space of a matrix and how it relates to solving inhomogeneous equations and determining whether a transformation is one-to-one.

Transcript

Let's say I have some matrix A. If I'm trying to determine the null space of A, I'm essentially asking -- I'm essentially just asking, look if we set up the equation Ax is equal to the 0 vector, the null space of A is all the x's that satisfy this equation. All the x's that satisfy that equation. Ax is equal to the 0 vector, or you could call it th... Read More

Key Insights

• ☺️ The null space of a matrix A consists of all the solutions to the equation Ax = 0.
• 🥋 The null space can be found by putting the augmented matrix with A and the zero vector into reduced row echelon form and un-augmenting it.
• 🛰️ The solution set to an inhomogeneous equation Ax = b includes a particular solution plus any vector multiplied by a scalar from the null space.
• 👾 For a transformation to be one-to-one, the null space of its transformation matrix must be trivial, meaning it only contains the zero vector.
• 👾 The trivial null space implies that the columns of the matrix are linearly independent and form a basis for the column space.

Questions & Answers

Q: What is the null space of a matrix?

The null space of a matrix A is all the solutions to the equation Ax = 0. It is the set of vectors that get mapped to the zero vector by the transformation represented by A.

Q: How can you find the null space?

To find the null space, you can set up an augmented matrix with A and the zero vector on the right side. Then, perform row operations to put the left side in reduced row echelon form. The vectors obtained from the system when un-augmented form the basis for the null space.

Q: How does the null space relate to solving inhomogeneous equations?

When solving the inhomogeneous equation Ax = b, the null space becomes important. The solution set will consist of a particular solution plus any vector multiplied by a scalar from the null space. This is because adding any vector from the null space to the particular solution still satisfies the equation.

Q: What is the significance of the null space for the one-to-one property of a transformation?

The null space being trivial, meaning it only contains the zero vector, is a condition for a transformation to be one-to-one. If the null space is trivial, then any solution to the inhomogeneous equation Ax = b will only have one particular solution, implying a one-to-one mapping.

Summary & Key Takeaways

• The null space of a matrix A is all the solutions to the equation Ax = 0.

• To find the null space, you can set up an augmented matrix with A and the zero vector, perform row operations to put it in reduced row echelon form, and then un-augment it to create the system.

• The null space is spanned by the vectors obtained from the system, and any solution to the system can be expressed as a linear combination of these vectors.

• If you're solving an inhomogeneous equation Ax = b, there will be a particular solution plus a member of the null space in the solution set.