## 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 $N(A)$?

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If $A \in \mathbb{R}^{m \times n}$ and its transpose $A^T \in \mathbb{R}^{n \times m}$, what is $R(A^T)$?

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According to the property $\dim(N(A)) + \dim(R(A)) = n$, what does this imply?

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## 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

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## 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.