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Chapter 5: Problem 9

If \(A\) is an \(m \times n\) matrix of rank \(r,\) what are the dimensions of \(N(A)\) and \(N\left(A^{T}\right) ?\) Explain.

Short Answer

Expert verified
The dimensions of the null space of matrix A (N(A)) are \( n - r \), and the dimensions of the null space of the transpose of matrix A (N(\(A^T\))) are \( m - r \).

Step by step solution

01

Remember Rank-Nullity Theorem

The Rank-Nullity Theorem states that for any matrix A with dimensions m 脳 n, the sum of the rank of A (denoted as rank(A)) and the dimension of the null space of A (denoted as nullity(A)) is equal to the number of columns of A (n). Mathematically, this is expressed as: rank(A) + nullity(A) = n
02

Find Dimensions of Null Space of A (N(A))

We are given rank(A) = r. Now using the Rank-Nullity theorem formula for matrix A, we can find the dimension of the null space of A (nullity(A)): r + nullity(A) = n Thus, nullity(A) = n - r Hence, the dimensions of the null space of A (N(A)) is nullity(A) = n - r.
03

Remember Rank-Nullity Theorem for Transpose Matrix

The Rank-Nullity theorem also applies to the transpose of a matrix. If A^T is the transpose of the A matrix with dimensions n 脳 m, then, the sum of the rank of A^T (denoted as rank(A^T)) and the dimension of the null space of A^T (denoted as nullity(A^T)) is equal to the number of columns of A^T (m). Mathematically, this is expressed as: rank(A^T) + nullity(A^T) = m
04

Find Dimensions of Null Space of A Transpose (N(A^T))

Since the rank of a matrix and its transpose are equal (rank(A) = rank(A^T)), we have rank(A^T) = r. Now using the Rank-Nullity theorem formula for matrix A^T, we can find the dimension of the null space of A^T (nullity(A^T)): r + nullity(A^T) = m Thus, nullity(A^T) = m - r Hence, the dimensions of the null space of A transpose (N(A^T)) are nullity(A^T) = m - r. In conclusion, the dimensions of the null space of matrix A (N(A)) are n - r, and the dimensions of the null space of the transpose of matrix A (N(A^T)) are m - r.

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Most popular questions from this chapter

Sketch the set of points \(\left(x_{1}, x_{2}\right)=\mathbf{x}^{T}\) in \(\mathbb{R}^{2}\) such that (a) \(\|\mathbf{x}\|_{2}=1\) (b) \(\|\mathbf{x}\|_{1}=1\) (c) \(\|\mathbf{x}\|_{\infty}=1\)

Let \(A\) be an \(m \times n\) matrix of rank \(n\) and let \(P=A\left(A^{T} A\right)^{-1} A^{T}\) (a) Show that \(P \mathbf{b}=\mathbf{b}\) for every \(\mathbf{b} \in R(A) .\) Ex plain this property in terms of projections. (b) If \(\mathbf{b} \in R(A)^{\perp},\) show that \(P \mathbf{b}=\mathbf{0}\) (c) Give a geometric illustration of parts and (b) if \(R(A)\) is a plane through the origin in \(\mathbb{R}^{3}\)

Show that if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in an inner product space that satisfy the Pythagorean law $$\|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}$$ then \(\mathbf{u}\) and \(\mathbf{v}\) must be orthogonal.

Show that if \(P\) is a symmetric permutation matrix, then \(P^{2 k}=I\) and \(P^{2 k+1}=P\)

Prove that the transpose of an orthogonal matrix is an orthogonal matrix.
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