# Tagged: rank-nullity theorem

## Problem 713

Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when
$A= \begin{bmatrix} 1 & 2 & 1 \\ 1 & 1 & 3 \\ 0 & 0 & 0 \end{bmatrix} .$ Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.

## Problem 712

Let $A$ be an $m \times n$ matrix.
Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.

## Problem 676

Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by
$T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = 2a + (b-d)x – (a+c)x^2 + (a+b-c-d)x^3.$

Find the rank and nullity of $T$.

## Problem 605

Let $T:\R^2 \to \R^3$ be a linear transformation such that
$T\left(\, \begin{bmatrix} 3 \\ 2 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 4\\ 3 \end{bmatrix} \,\right) =\begin{bmatrix} 0 \\ -5 \\ 1 \end{bmatrix}.$

(a) Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).

(b) Determine the rank and nullity of $T$.

(The Ohio State University, Linear Algebra Midterm)

## Problem 541

Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.
Consider a linear transformation $T:U\to V$.

Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).

## Problem 484

Let $A$ be a square matrix and its characteristic polynomial is given by
$p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$.

(The Ohio State University, Linear Algebra Final Exam Problem)

## Problem 429

Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

## Problem 379

Find all the eigenvalues and eigenvectors of the matrix
$A=\begin{bmatrix} 3 & 9 & 9 & 9 \\ 9 &3 & 9 & 9 \\ 9 & 9 & 3 & 9 \\ 9 & 9 & 9 & 3 \end{bmatrix}.$

(Harvard University, Linear Algebra Final Exam Problem)

## Problem 376

(a) Let
$A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.$ Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.

(b) Let
$A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.$ One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.

## Problem 370

Let $T: \R^2 \to \R^2$ be a linear transformation such that
$T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\ 1 \end{bmatrix}, T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 3 \\ 2 \end{bmatrix}.$ Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 369

Let $T:\R^3 \to \R^2$ be a linear transformation such that
$T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 0 \\ 1 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 1 \\ 0 \end{bmatrix},$ where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
Then find the rank and the nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 366

Let $A=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}$.

(a) Find an orthonormal basis of the null space of $A$.

(b) Find the rank of $A$.

(c) Find an orthonormal basis of the row space of $A$.

(The Ohio State University, Linear Algebra Exam Problem)

## Problem 352

A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors
$\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\in \R^n$ satisfying the linear equation of the form
$a_1x_1+a_2x_2+\cdots+a_nx_n=b,$ where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.
Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation
$a_1x_1+a_2x_2+\cdots+a_nx_n=0,$ where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.
(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

## Problem 329

Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.

(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.

(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.
Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then
$B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}$ is a basis of $\R^n$.

(c) Each vector $\mathbf{u}\in \R^n$ can be expressed as
$\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}$ for some vector $\mathbf{v}\in \calN(T)$.

## Problem 320

(a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.

(b) Find the rank and nullity of the matrix $A$ in part (a).

## Problem 303

Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
$\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}, \text{ and } \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}.$ Then find the rank of the matrix $A$.

(Purdue University, Linear Algebra Final Exam Problem)

## Problem 276

Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
$T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix} a_0+a_2 & -a_0+a_3\\ a_1-a_2 & -a_1-a_3 \end{bmatrix}$ for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.

## Problem 217

Let $A, B, C$ are $2\times 2$ diagonalizable matrices.

The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.

From this information, determine the rank of the matrices $A, B,$ and $C$.

Graphs of characteristic polynomials

## Problem 164

Let $T:\R^4 \to \R^3$ be a linear transformation defined by
$T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.$

(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.

(b) Find a basis for the null space of $T$.

(c) Find the rank of the linear transformation $T$.

(The Ohio State University Linear Algebra Exam Problem)

## Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$
Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$.
(a) Show that $T$ is a linear transformation.
(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.
(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.