Category: Linear Algebra

10/23/2016

Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$

Problem 156

Let $T: \R^3 \to \R^2$ be a linear transformation such that
\[T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
4
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
2 \\
5
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
3 \\
6
\end{bmatrix},\] where
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}, \mathbf{e}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix}\] are the standard unit basis vectors of $\R^3$.
For any vector $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.

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10/22/2016

Linear Properties of Matrix Multiplication and the Null Space of a Matrix

Problem 155

Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.

Then find $A\mathbf{w}$.

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10/21/2016

Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$

Problem 154

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 $T$.

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10/20/2016

Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis

Problem 153

Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient.
Let $W$ be the following subset of $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and $p^{\prime\prime}(x)$ is the second derivative of $p(x)$.

Show that $W$ is a subspace of $P_3$ and find a basis for $W$.

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10/20/2016

Find a Basis for a Subspace of the Vector Space of $2\times 2$ Matrices

Problem 152

Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where
\begin{align*}
A_1=\begin{bmatrix}
1 & 2 \\
-1 & 3
\end{bmatrix}, \quad
A_2=\begin{bmatrix}
0 & -1 \\
1 & 4
\end{bmatrix}, \quad
A_3=\begin{bmatrix}
-1 & 0 \\
1 & -10
\end{bmatrix}, \quad
A_4=\begin{bmatrix}
3 & 7 \\
-2 & 6
\end{bmatrix}.
\end{align*}
Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.

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10/19/2016

Any Vector is a Linear Combination of Basis Vectors Uniquely

Problem 151

Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a basis for a vector space $V$ over a scalar field $K$. Then show that any vector $\mathbf{v}\in V$ can be written uniquely as
\[\mathbf{v}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+c_3\mathbf{v}_3,\] where $c_1, c_2, c_3$ are scalars.

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10/19/2016

A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors

Problem 150

Show that the set
\[S=\{1, 1-x, 3+4x+x^2\}\] is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.

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10/16/2016

Nilpotent Matrices and Non-Singularity of Such Matrices

Problem 146

Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.

Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

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10/14/2016

Subspaces of Symmetric, Skew-Symmetric Matrices

Problem 143

Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

(a) The set $S$ consisting of all $n\times n$ symmetric matrices.

(b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

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10/13/2016

Find a Value of a Linear Transformation From $\R^2$ to $\R^3$

Problem 142

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}$ are unit vectors of $\R^2$ and
\[\mathbf{u}_1= \begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}, \quad \mathbf{u}_2=\begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}.\] Then find $T\left(\begin{bmatrix}
3 \\
-2
\end{bmatrix}\right)$.

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10/11/2016

Linear Independent Vectors and the Vector Space Spanned By Them

Problem 141

Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let $\mathbf{u}_{n+1}\in V$. Show that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent if and only if $\mathbf{u}_{n+1} \not \in U$.

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10/10/2016

Rank and Nullity of a Matrix, Nullity of Transpose

Problem 140

Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.

(a) $\calN(A)=\calN(A^{\trans}A)$.

(b) $\rk(A)=\rk(A^{\trans}A)$.

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10/09/2016

Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero

Problem 139

Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
\[A_1^2+A_2^2+\cdots+A_m^2=\calO,\] where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.

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10/09/2016

How to Find the Determinant of the $3\times 3$ Matrix

Problem 138

Find the determinant of the matix
\[A=\begin{bmatrix}
100 & 101 & 102 \\
101 &102 &103 \\
102 & 103 & 104
\end{bmatrix}.\]

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10/08/2016

Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less

Problem 137

Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let
\[U=\{ p(x) \in P_n(\R) \mid p(1)=0\}\] be a subspace of $P_n(\R)$.

Find a basis for $U$ and determine the dimension of $U$.

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10/07/2016

Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)

Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

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10/06/2016

Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$

Problem 135

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.

(a) $\rk(AB) \leq \rk(A)$.

(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.

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10/05/2016

Subspaces of the Vector Space of All Real Valued Function on the Interval

Problem 134

Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.

(a) $S=\{f(x) \in V \mid f(0)=f(1)\}$.

(b) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.

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10/04/2016

Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?

Problem 133

Find a square root of the matrix
\[A=\begin{bmatrix}
1 & 3 & -3 \\
0 &4 &5 \\
0 & 0 & 9
\end{bmatrix}.\]

How many square roots does this matrix have?

(University of California, Berkeley Qualifying Exam)

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10/03/2016

Find a Basis For the Null Space of a Given $2\times 3$ Matrix

Problem 132

Let
\[A=\begin{bmatrix}
1 & 1 & 0 \\
1 &1 &0
\end{bmatrix}\] be a matrix.

Find a basis of the null space of the matrix $A$.

(Remark: a null space is also called a kernel.)

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