Bases and Coordinate Vectors

Bases and Coordinate Vectors

Definition

Let $V$ be a vector space over a scalar field $K$.

  1. A basis $B$ for $V$ is a linearly independent spanning set for $V$.
  2. Suppose that $B=\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ is a basis for $V$. Let $\mathbf{v}\in V$ and write it as $\mathbf{v}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n$, where $c_1, \dots, c_n\in K$. Then the coordinate vector of $\mathbf{v}$ with respect to the basis $B$ is
    \[[\mathbf{v}]_B=\begin{bmatrix}
    c_1 \\
    \vdots \\
    c_n
    \end{bmatrix} \in \R^n.\]
Summary

Let $V$ be a vector space over a scalar field $K$. Suppose that $B=\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ is a basis for $V$. Let $S=\{\mathbf{w}_1, \dots, \mathbf{w}_k\}$ be a set of vectors in $V$. Let $T=\{[\mathbf{w}_1]_B, \dots, [\mathbf{w}_k]_B\}$ be the set of the coordinate vectors of $S$.

  1. Any vector $\mathbf{v}$ can be uniquely written as $\mathbf{v}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n$, where $c_1, \dots, c_n\in K$.
  2. The dimension of the coordinate vector $[\mathbf{v}]_B$ is the dimension of the vector space $V$.
  3. The set $S$ is linearly independent if and only if $T$ is linearly independent.
  4. $S$ is a basis for $\Span(S)$ if and only if $T$ is a basis for $\Span(T)$ in $\R^n$.

=solution

Problems

  1. 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.

  2. 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.

  3. 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 $\Span(S)$.

  4. Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let
    \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.
    (The Ohio State University)

  5. 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.

  6. Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where
    \begin{align*}
    p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\
    p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.
    \end{align*}
    (a) Find a basis of $P_2$ among the vectors of $S$.
    (b) Let $B’$ be the basis you obtained in part (a). For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.
    (The Ohio State University)

  7. 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)=1, &p_2(x)=x^2+x+1, \\
    &p_3(x)=2x^2, &p_4(x)=x^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.
    (The Ohio State University)

  8. Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less.
    Let $S=\{p_1(x), p_2(x), p_3(x)\}$, where
    \[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.\] (a) Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for $P_2$.
    (b) Find the coordinate vector of $p(x)=x^2+2x+3\in P_2$ with respect to the basis $S$.

  9. Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.
    Let
    \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where
    \begin{align*}
    p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\
    p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.
    \end{align*}
    (a) Find a basis $Q$ of the span $\Span(S)$ consisting of polynomials in $S$.
    (b) For each polynomial in $S$ that is not in $Q$, find the coordinate vector with respect to the basis $Q$.
    (The Ohio State University)

  10. Let $V$ be the vector space of all $2\times 2$ real matrices. Let $S=\{A_1, A_2, A_3, A_4\}$, where
    \[A_1=\begin{bmatrix}
    1 & 2\\
    -1& 3
    \end{bmatrix}, A_2=\begin{bmatrix}
    0 & -1\\
    1& 4
    \end{bmatrix}, A_3=\begin{bmatrix}
    -1 & 0\\
    1& -10
    \end{bmatrix}, A_4=\begin{bmatrix}
    3 & 7\\
    -2& 6
    \end{bmatrix}.\] Then find a basis for the span $\Span(S)$.

  11. Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
    \[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\] be a subset in $C[-1, 1]$.
    (a) Prove that $V$ is a subspace of $C[-1, 1]$.
    (b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.
    (c) Prove that $B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$.

  12. 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 $U$.

  13. 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 $S$.
    \[[v_1]_B=\begin{bmatrix}
    1 \\
    0 \\
    0 \\
    0
    \end{bmatrix}, [v_2]_B=\begin{bmatrix}
    0 \\
    1 \\
    0 \\
    0
    \end{bmatrix}, [v_3]_B=\begin{bmatrix}
    1 \\
    1 \\
    0 \\
    0
    \end{bmatrix}.\]
  14. Let $V$ be a vector space and $B$ be a basis for $V$.
    Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.
    Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$. After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form $\begin{bmatrix}
    1 & 0 & 2 & 1 & 0 \\
    0 & 1 & 3 & 0 & 1 \\
    0 & 0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 & 0
    \end{bmatrix}$.
    (a) What is the dimension of $V$?
    (b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$?
    (The Ohio State University)

  15. Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero. That is,
    \begin{equation*}
    V:=\left\{ A=\begin{bmatrix}
    a_{11} & 0 & \dots & 0 \\
    0 &a_{22} & \dots & 0 \\
    0 & 0 & \ddots & \vdots \\
    0 & 0 & \dots & a_{nn}
    \end{bmatrix} \quad \middle| \quad
    \begin{array}{l}
    a_{11}, \dots, a_{nn} \in \C,\\
    \tr(A)=0 \\
    \end{array}
    \right\}
    \end{equation*}
    Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.
    (a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)
    (b) Show that matrices
    \[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\] are a basis for the vector space $V$.
    (c) Find the dimension of $V$.

  16. Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$. Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
    (a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.
    (b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.