Linear Algebra: Range and Null Space
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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 $N(A)$?

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

    <p>The span of the columns of $A^T$.</p> Signup and view all the answers

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

    <p>The dimension of the null space and range space of $A$ add up to $n$.</p> 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.

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