Bases and coordinate systems

We will look at some examples of bases in this activity.

1. In the preview activity, we worked with the set of vectors in $\real^2$: . Explain why these vectors form a basis for $\real^2$.

2. Consider the set of vectors in $\real^3$ and determine whether they form a basis for $\real^3$.

3. Do the vectors form a basis for $\real^3$?

4. Explain why the vectors $\evec_1,\evec_2,\evec_3$ form a basis for $\real^3$.

5. If a set of vectors $\vvec_1,\vvec_2,\ldots,\vvec_n$ forms a basis for $\real^m$, what can you guarantee about…

Let's begin with the basis $\bcal = \{\vvec_1,\vvec_2\}$ of $\real^2$ where .

1. If the coordinates of $\xvec$ in the basis $\bcal$ are $\coords{\xvec}{\bcal} = \twovec{-2}{4}$, what is the vector $\xvec$?

2. If $\xvec = \twovec{3}{5}$, find the coordinates of $\xvec$ in the basis $\bcal$; that is, find $\coords{\xvec}{\bcal}$.

3. Find a matrix $A$ such that, for any vector $\xvec$, we have $\xvec = A\coords{\xvec}{\bcal}$. Explain why this matrix is invertible.

4. Using what you found in the previous part, find a matrix $B$ such that, for any vector $\xvec$, we have $\coords{\xvec}{\bcal} = B\xvec$. What is the relationship between th…

Consider vectors and let $\bcal = \{\vvec_1,\vvec_2\}$ and $\ccal = \{\wvec_1,\wvec_2\}$.

1. Explain why $\bcal$ and $\ccal$ are both bases of $\real^2$.

2. If $\xvec = \twovec{5}{-3}$, find $\coords{\xvec}{\bcal}$ and $\coords{\xvec}{\ccal}$.

3. If $\coords{\xvec}{\bcal}=\twovec{2}{-4}$, find $\xvec$ and $\coords{\xvec}{\ccal}$.

4. If $\coords{\xvec}{\ccal}=\twovec{-3}{2}$, find $\xvec$ and $\coords{\xvec}{\bcal}$.

5. Find a matrix $Q$ such that $\coords{\xvec}{\bcal} = Q\coords{\xvec}{\ccal}$.

Consider the following vectors in $\real^4$: .

1. Explain why $\bcal=\{\vvec_1,\vvec_2,\vvec_3,\vvec_4\}$ forms a basis for $\real^4$.

2. Explain how to convert $\coords{\xvec}{\bcal}$, the representation of a vector $\xvec$ in the coordinates defined by $\bcal$, into $\xvec$, its representation in the standard coordinate system.

3. Explain how to convert the vector $\xvec$ into $\coords{\xvec}{\bcal}$, its representation in the coordinate system defined by $\bcal$.

4. If $\xvec=\fourvec{23}{12}{10}{19}$, find $\coords{\xvec}{\bcal}$.

5. If $\coords{\xvec}{\bcal}=\fourvec{3}{1}{-3}{-4}$, find $\xvec$.

Consider the following vectors in $\real^3$: .

1. Do these vectors form a basis for $\real^3$? Explain your thinking.

2. Find a subset of these vectors that forms a basis of $\real^3$.

3. Suppose you have a set of vectors $\vvec_1, \vvec_2,\ldots,\vvec_6$ in $\real^4$ such that . Find a subset of the vectors that forms a basis for $\real^4$.

This exercise involves a simple Fourier transform, which will play an important role in the next section.

Suppose that we have the vectors .

1. Explain why $\bcal=\{\vvec_1,\vvec_2,\vvec_3\}$ is a basis for $\real^3$. Notice that you may enter $\cos\left(\frac\pi6\right)$ into Sage as cos(pi/6).

2. If $\xvec=\threevec{15}{15}{15}$, find $\coords{\xvec}{\bcal}$.

3. Find the matrices $P_{\bcal}$ and $P_{\bcal}^{-1}$. If $\xvec=\threevec{x_1}{x_2}{x_3}$ and $\coords{\xvec}{\bcal} = \threevec{c_1}{c_2}{c_3}$, explain why $c_1$ is the average of $x_1$, $x_2$, and $x_3$.

Determine whether the following statements are true or false and provide a justification for your response.

1. If the columns of a matrix $A$ form a basis for $\real^m$, then $A$ is invertible.

2. There must be 125 vectors in a basis for $\real^{125}$.

3. If $\bcal=\{\vvec_1,\vvec_2,\ldots,\vvec_n\}$ is a basis of $\real^m$, then every vector in $\real^m$ can be expressed as a linear combination of basis vectors.

4. The coordinates $\coords{\xvec}{\bcal}$ are the weights that form $\xvec$ as a linear combination of basi…

Provide a justification for your response to each of the following questions.

1. Suppose you have $m$ linearly independent vectors in $\real^m$. Can you guarantee that they form a basis of $\real^m$?

2. If $A$ is an invertible $m\times m$ matrix, do the columns necessarily form a basis of $\real^m$?

3. Suppose we have an invertible $m\times m$ matrix $A$, and we perform a sequence of row operations on $A$ to form a matrix $B$. Can you guarantee that the columns…

Suppose that $A=\left[\begin{array}{rr} 2 & 1 \\ 1& 2 \\ \end{array}\right]$ and .

1. Explain why $\bcal=\{\vvec_1,\vvec_2\}$ is a basis for $\real^2$.

2. Find $A\vvec_1$ and $A\vvec_2$.

3. Use what you found in the previous part of this problem to find $\coords{A\vvec_1}{\bcal}$ and $\coords{A\vvec_2}{\bcal}$.

4. If $\coords{\xvec}{\bcal} = \twovec{1}{-5}$, find $\coords{A\xvec}{\bcal}$.

5. Find a matrix $D$ such that $\coords{A\xvec}{\bcal} = D\coords{\xvec}{\bcal}$.

You should find that the matrix $D$ is a very simple matrix, which means that this basis $\bcal$ is well suited to study the effect of multiplication by $A$. This observation is the central idea of the next chapter.