6 Questions
What is the set of all vectors in $\mathbb{R}^{m}$ that can be written as linear combinations of the columns of $A$?
$R(A)$
Which of the following statements is true about the rank of matrix $A$?
The rank is denoted by $r$ and $r \leq \min(m, n)$.
If $A \in \mathbb{R}^{m \times n}$, what is the relationship between $N(A)$ and $\mathbb{R}^{n}$?
$N(A)$ is a subset of $\mathbb{R}^{n}$.
Which of the following correctly describes $N(A)$?
It is the set of all vectors $x \in \mathbb{R}^{n}$ such that $Ax = 0$.
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 $A^T$.
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 $n$.
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
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.
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