Podcast Beta
Questions and Answers
What is the set of all vectors in $\mathbb{R}^{m}$ that can be written as linear combinations of the columns of $A$?
Which of the following statements is true about the rank of matrix $A$?
If $A \in \mathbb{R}^{m \times n}$, what is the relationship between $N(A)$ and $\mathbb{R}^{n}$?
Which of the following correctly describes $N(A)$?
Signup and view all the answers
If $A \in \mathbb{R}^{m \times n}$ and its transpose $A^T \in \mathbb{R}^{n \times m}$, what is $R(A^T)$?
Signup and view all the answers
According to the property $\dim(N(A)) + \dim(R(A)) = n$, what does this imply?
Signup and view all the answers
Study Notes
Range Space, R(A)
- The range space of a matrix A is the set of all vectors in R^m that can be written as linear combinations of the columns of A.
- Formally, R(A) = span({a_1, a_2,..., a_n}) = {y = Ax, x ∈ R^n, y ∈ R^m}
- Similarly, R(A^T) = span({a_1, a_2,..., a_m}) = {w = A^Tz, z ∈ R^m, w ∈ R^n}
Properties of Range Space
- R(A) ⊂ R^m
- dim(R(A)) or dim(R(A^T)) is the rank of A, denoted by r, and r ≤ min(m, n)
Null Space, N(A)
- The null space of a matrix A is the set of all vectors x mapped into zero by A.
- N(A) = {x: Ax = 0, x ∈ R^n}
- Similarly, N(A^T) = {z: Az = 0, z ∈ R^m}
Properties of Null Space
- N(A) ⊂ R^n
- dim(N(A)) + dim(R(A)) = n
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers the concepts of range and null space in linear algebra, including the definition of range space, null space, and their properties. It's a crucial topic in understanding linear transformations and vector spaces.