Definition

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

- A basis $B$ for $V$ is a linearly independent spanning set for $V$.
- 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 $S$.

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

=solution

### Problems

- Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.
- 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. - 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. - 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 $\Span(S)$. - 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*) - Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis

\[B = \left\{ 1+x , 1+x^2 , x – x^2 + 2x^3 , 1 – x – x^2 \right\}.\]Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis $B$. - 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. - 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*) - 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*) - 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 $S$. - 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)* - Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less.
**(a)**Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$.**(b)**Write the polynomial $f(x) = 2 + 3x – x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 \}$. - 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 $\Span(S)$. - 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 $V$. - 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 $U$. - 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}.\] - 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)* - 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 $V$. - 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 $W$. - Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$. Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible. Prove that for each vector $\mathbf{v} \in V$, the vector $S^{-1}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to the basis $B$.

Linear AlgebraVersion 0 (11/15/2017)

- Introduction to Matrices
- Elementary Row Operations
- Gaussian-Jordan Elimination
- Solutions of Systems of Linear Equations
- Linear Combination and Linear Independence
- Nonsingular Matrices
- Inverse Matrices
- Subspaces in $\R^n$
- Bases and Dimension of Subspaces in $\R^n$
- General Vector Spaces
- Subspaces in General Vector Spaces
- Linearly Independency of General Vectors
- Bases and Coordinate Vectors
- Dimensions of General Vector Spaces
- Linear Transformation from $\R^n$ to $\R^m$
- Linear Transformation Between Vector Spaces
- Orthogonal Bases
- Determinants of Matrices
- Computations of Determinants
- Introduction to Eigenvalues and Eigenvectors
- Eigenvectors and Eigenspaces
- Diagonalization of Matrices
- The Cayley-Hamilton Theorem
- Dot Products and Length of Vectors
- Eigenvalues and Eigenvectors of Linear Transformations
- Jordan Canonical Form