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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 $A$?
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 $A$?
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 $\mathbb{R}^{n}$?
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 $N(A)$?
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 $R(A^T)$?
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?
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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