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Linear Algebra: Range and Null Space

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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 $A$?

$R(A)$ (correct)

$\mathbb{R}^{n}$

$N(A)$

$N(A^T)$

Which of the following statements is true about the rank of matrix $A$?

The rank is denoted by $r$ and $r \geq \max(m, n)$.

The rank is always equal to $n$.

The rank is always equal to $m$.

The rank is denoted by $r$ and $r \leq \min(m, n)$. (correct)

If $A \in \mathbb{R}^{m \times n}$, what is the relationship between $N(A)$ and $\mathbb{R}^{n}$?

$N(A)$ is equal to $\mathbb{R}^{m}$.

$N(A)$ is a subset of $\mathbb{R}^{m}$.

$N(A)$ is equal to $\mathbb{R}^{n}$.

$N(A)$ is a subset of $\mathbb{R}^{n}$. (correct)

Which of the following correctly describes $N(A)$?

It is the set of all vectors $x \in \mathbb{R}^{n}$ such that $Ax = 0$. 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 $R(A^T)$?

The span of the columns of $A^T$. Signup and view all the answers

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

The dimension of the null space and range space of $A$ add up to $n$. 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

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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.