Tagged: null space

Find a Basis for Nullspace, Row Space, and Range of a Matrix

Problem 704

Let $A=\begin{bmatrix}
2 & 4 & 6 & 8 \\
1 &3 & 0 & 5 \\
1 & 1 & 6 & 3
\end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.

(b) Find a basis for the row space of $A$.

(c) Find a basis for the range of $A$ that consists of column vectors of $A$.

(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

 
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Find the Dimension of the Subspace of Vectors Perpendicular to Given Vectors

Problem 578

Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
\[\mathbf{a}=\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix}
0 \\
1 \\
-1 \\
0
\end{bmatrix}.\]

Namely,
\[V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.\]

(a) Prove that $V$ is a subspace of $\R^4$.

(b) Find a basis of $V$.

(c) Determine the dimension of $V$.

 
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Determine Whether Given Subsets in $\R^4$ are Subspaces or Not

Problem 480

(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(The Ohio State University, Linear Algebra Final Exam Problem)
 
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Null Space, Nullity, Range, Rank of a Projection Linear Transformation

Problem 450

Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]

(a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.

(b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.

(c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.

(d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.

(e) Let
\[B=\left\{\, \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
-1 \\
1 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
-1 \\
1
\end{bmatrix} \,\right\}\] be a basis for $\R^3$.
Calculate the coordinates of $\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}$ with respect to $B$.

(The Ohio State University, Linear Algebra Exam Problem)

 
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Subspace Spanned By Cosine and Sine Functions

Problem 435

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\] We put
\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]

(a) Prove that the map $f$ is a linear transformation.

(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)

(d) Define a map $g:V \to V$ by
\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\] Prove that the map $g$ is a linear transformation.

(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

 
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Find All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix

Problem 400

Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
10001 & 3 & 5 & 7 &9 & 11 \\
1 & 10003 & 5 & 7 & 9 & 11 \\
1 & 3 & 10005 & 7 & 9 & 11 \\
1 & 3 & 5 & 10007 & 9 & 11 \\
1 &3 & 5 & 7 & 10009 & 11 \\
1 &3 & 5 & 7 & 9 & 10011
\end{bmatrix}.\]

(MIT, Linear Algebra Homework Problem)
 
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If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution

Problem 395

Suppose that the vectors
\[\mathbf{v}_1=\begin{bmatrix}
-2 \\
1 \\
0 \\
0 \\
0
\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
-4 \\
0 \\
-3 \\
-2 \\
1
\end{bmatrix}\] are a basis vectors for the null space of a $4\times 5$ matrix $A$. Find a vector $\mathbf{x}$ such that
\[\mathbf{x}\neq0, \quad \mathbf{x}\neq \mathbf{v}_1, \quad \mathbf{x}\neq \mathbf{v}_2,\] and
\[A\mathbf{x}=\mathbf{0}.\]

(Stanford University, Linear Algebra Exam Problem)
 
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Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix

Problem 384

Let $A$ be an $n\times n$ matrix with the characteristic polynomial
\[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable.

(a) Find the size of the matrix $A$.

(b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.

(c) Find the nullity of $A$.

(The Ohio State University, Linear Algebra Final Exam Problem)
 
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Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities

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 $\lambda=0$.

 
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Hyperplane in $n$-Dimensional Space Through Origin is a Subspace

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 $n-1$.

 
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Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space

Problem 313

(a) Let $A=\begin{bmatrix}
1 & 2 & 1 \\
3 &6 &4
\end{bmatrix}$ and let
\[\mathbf{a}=\begin{bmatrix}
-3 \\
1 \\
1
\end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}
-2 \\
1 \\
0
\end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix}
1 \\
1
\end{bmatrix}.\] For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.

(b) Find a basis of the null space of the matrix $B=\begin{bmatrix}
1 & 1 & 2 \\
-2 &-2 &-4
\end{bmatrix}$.

 
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