Tagged: basis of a vector space

Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.

Problem 574

Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.

(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.

(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.

 

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The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization

Problem 539

Consider the $2\times 2$ real matrix
\[A=\begin{bmatrix}
1 & 1\\
1& 3
\end{bmatrix}.\]

(a) Prove that the matrix $A$ is positive definite.

(b) Since $A$ is positive definite by part (a), the formula
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans} A \mathbf{y}\] for $\mathbf{x}, \mathbf{y} \in \R^2$ defines an inner product on $\R^n$.
Consider $\R^2$ as an inner product space with this inner product.

Prove that the unit vectors
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}\] are not orthogonal in the inner product space $\R^2$.

(c) Find an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ of $\R^2$ from the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ using the Gram-Schmidt orthogonalization process.

 

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Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices

Problem 375

Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ elements.
Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

 

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Coordinate Vectors and Dimension of Subspaces (Span)

Problem 350

Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.
Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.
\[[v_1]_B=\begin{bmatrix}
1 \\
0 \\
0 \\
0
\end{bmatrix}, [v_2]_B=\begin{bmatrix}
0 \\
1 \\
0 \\
0
\end{bmatrix}, [v_3]_B=\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}.\]

 

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Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix

Problem 287

Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.

Determine which matrices are in the subspace $W$ and find the dimension of $W$.

(a) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &b &0 \\
0 & 0 & c
\end{bmatrix},\] where $a, b, c$ are distinct real numbers.

(b) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &a &0 \\
0 & 0 & b
\end{bmatrix},\] where $a, b$ are distinct real numbers.

 

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Prove a Given Subset is a Subspace and Find a Basis and Dimension

Problem 270

Let
\[A=\begin{bmatrix}
4 & 1\\
3& 2
\end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.
\[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\]

(a) Prove that the subset $V$ is a subspace of $\R^2$.

(b) Find a basis for $V$ and determine the dimension of $V$.

 

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Basis and Dimension of the Subspace of All Polynomials of Degree 4 or Less Satisfying Some Conditions.

Problem 256

Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients.
Let $W$ be the subspace of $P_2$ by
\[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of $W$.

 

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Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$

Problem 255

Let $B=\{\mathbf{v}_1, \mathbf{v}_2 \}$ be a basis for the vector space $\R^2$, and let $T:\R^2 \to \R^2$ be a linear transformation such that
\[T(\mathbf{v}_1)=\begin{bmatrix}
1 \\
-2
\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}
3 \\
1
\end{bmatrix}.\]

If $\mathbf{e}_1=\mathbf{v}_1+2\mathbf{v}_2 \text{ and } \mathbf{e}_2=2\mathbf{v}_1-\mathbf{u}_2$, where $\mathbf{e}_1, \mathbf{e}_2$ are the standard unit vectors in $\R^2$, then find the matrix of $T$ with respect to the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$.
 

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Linear Transformation and a Basis of the Vector Space $\R^3$

Problem 182

Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$.
Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$.

Show that the vectors $\mathbf{x}, T\mathbf{x}, T^2\mathbf{x}$ form a basis for $\R^3$.

(The Ohio State University Linear Algebra Exam Problem)
 

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Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix

Problem 180

Suppose the following information is known about a $3\times 3$ matrix $A$.
\[A\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix}=6\begin{bmatrix}
1 \\
2 \\
1
\end{bmatrix},
\quad
A\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}, \quad
A\begin{bmatrix}
2 \\
-1 \\
0
\end{bmatrix}=3\begin{bmatrix}
1 \\
-1 \\
1
\end{bmatrix}.\]

(a) Find the eigenvalues of $A$.

(b) Find the corresponding eigenspaces.

(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?

(Johns Hopkins University Linear Algebra Exam)
 

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A Matrix Representation of a Linear Transformation and Related Subspaces

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.)
 

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Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known

Problem 159

Let $T: \R^2 \to \R^2$ be a linear transformation.
Let
\[
\mathbf{u}=\begin{bmatrix}
1 \\
2
\end{bmatrix}, \mathbf{v}=\begin{bmatrix}
3 \\
5
\end{bmatrix}\] be 2-dimensional vectors.
Suppose that
\begin{align*}
T(\mathbf{u})&=T\left( \begin{bmatrix}
1 \\
2
\end{bmatrix} \right)=\begin{bmatrix}
-3 \\
5
\end{bmatrix},\\
T(\mathbf{v})&=T\left(\begin{bmatrix}
3 \\
5
\end{bmatrix}\right)=\begin{bmatrix}
7 \\
1
\end{bmatrix}.
\end{align*}
Let $\mathbf{w}=\begin{bmatrix}
x \\
y
\end{bmatrix}\in \R^2$.
Find the formula for $T(\mathbf{w})$ in terms of $x$ and $y$.

 

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Vector Space of Polynomials and Coordinate Vectors

Problem 157

Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where
\begin{align*}
&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\
&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.
\end{align*}
(a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.
(b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.
(c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.

 

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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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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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